Sharp pointwise estimates for analytic functions by the L p -norm of the real part

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "Sharp pointwise estimates for analytic functions by the L p -norm of the real part"

Transcript

1 Sharp pointwise estimates for analytic functions by the L p -norm of the real part Gershon Kresin and Vladimir Maz ya Abstract. We obtain sharp estimates of Re iα(z) (f(z) f(0))} by the L p -norm of Rf ω on the circle ζ = R, where z < R, 1 p, and α is a real valued function on D R. Here f is an analytic function in the disc D R = z : z < R} whose real part is continuous on D R, ω is a real constant, and Rf ω is orthogonal to some continuous function Φ on the circle ζ = R. We derive two types of estimates with vanishing and nonvanishing mean value of Φ. The cases Φ = 0 and Φ = 1 are discussed in more detail. In particular, we give explicit formulas for sharp constants in inequalities for Re iα(z) (f(z) f(0))} with p = 1,,. We also obtain estimates for f(z) f(0) in the class of analytic functions with two-sided bounds of argf(z) f(0)}. As a corollary, we find a sharp constant in the upper estimate of If(z) If(0) by Rf Rf(0) p which generalizes the classical Carathéodory-Plemelj estimate with p =. Keywords: Analytic functions, real part, pointwise estimates, orthogonality condition AMS: 30A10, 31A05, 31A0, 35J05 0 Introduction In the present paper we consider an analytic function f on the disk D R = z : z < R} whose real part is continuous on D R. We obtain sharp estimates for max z =r Reiα(z) (f(z) f(0))} by the L p -norm of Rf ω on the circle D R, where 0 r < R, 1 p, α is a real valued function on D R, and ω is a real constant. We assume that where Φ is a continuous function defined on the circle ζ = R. (Rf ω, Φ) = 0, (0.1) The Research Institute, The College of Judea and Samaria, Ariel, 44837, Israel Department of Mathematics, The Ohio State University, 31 West Eighteenth Avenue, Columbus, Ohio , USA 1

2 In Section 1 we prove the basic Lemma 1 which gives a general but somewhat implicit representation of the best constant C Φ, p (z, α(z)) in the estimate of Re iα(z) (f(z) f(0))} by Rf p with the orthogonality condition (0.1). Section concerns the inequality Re iα(z) (f(z) f(0))} C Φ, p (z, α(z)) Rf ω p (0.) with (Rf, Φ) = 0 and vanishing mean value of Φ for ζ = R. By ω in (0.) we mean an arbitrary real constant. The value Rf ω p in the right-hand side of (0.) can be replaced by E p (Rf) which stands for the best approximation of Rf by a real constant in the norm of L p ( D R ). The case Φ = 0 is treated in more details. As a corollary of Lemma 1 we find the representation for the sharp constant in (0.) C 0, p (z, α(z)) = R 1/p C 0, p (r/r, α(z)), (0.3) where C 0, p (γ, α) = γ π cos(ϕ α) γ cos α π 1 γ cos ϕ + γ p/(p 1) dϕ} (p 1)/p (0.4) if 1 < p <, and C 0, 1 (γ, α) = γ(1 + γ cos α ), (0.5) π(1 γ ) C 0, (γ, α) = 4 π sin α log γ sin α + (1 γ cos α) 1/ (1 γ ) 1/ + cos α arcsin (γ cos α) }. (0.6) In particular, we obtain the equality C 0, (γ, α) = γ[π(1 γ )] 1/. For Φ = 0, p = 1, and ω = A f (R) = maxrf(ζ) : ζ = R} inequality (0.) and formula (0.5) imply the Hadamard-Borel-Carathéodory inequality. Also note, that by (0.) and (0.6) with Φ = 0, p =, ω = 0, α = 0, and α = π/ one gets the estimates Rf(z) Rf(0) 4 π arcsin ( r R ) Rf, If(z) If(0) ( ) R + r π log Rf R r (see, for example, [8]). The first inequality is known as the Schwarz Arcussinus Formula. The right-hand side of the second inequality is, in fact, the sharp majorant for f(z) f(0). For Φ = 0, α = π/ and any z with z = r < R, inequality (0.) and formulas (0.3), (0.4) imply If(z) If(0) S p (r/r) Rf ω p (0.7)

3 with the sharp constant S p (γ) = κ(γ) 1 } (1 t ) (q 1)/ 1/q πr 1/p 1 [1 κ(γ)t] dt, (0.8) q where q = p/(p 1), κ(γ) = (γ)/(1 + γ ). The integral in the right-hand side of (0.8) can be expressed in terms of hypergeometric Gauss function and is evaluated explicitly for a some values of p. Section 3 concerns the inequality Re iα(z) (f(z) f(0))} C Φ, p (z, α(z)) Rf (Rf, Φ)/(1, Φ) p, which is valid by Lemma 1 for Φ with a nonvanishing mean value on the circle ζ = R. In case Φ = 1 the last inequality takes the form Re iα(z) (f(z) f(0))} C 1, p (z, α(z)) Rf Rf(0) p, (0.9) and the sharp constant is defined by C 1, p (z, α(z)) = R 1/p C 1, p (r/r, α(z)), where C 1, p (γ, α) = γ π min λ R π cos(ϕ α) γ cos α 1 γ cos ϕ + γ λ p/(p 1) dϕ} (p 1)/p (0.10) if 1 < p <, and C 1, 1 (γ, α) = γ π(1 γ ), (0.11) C 1, (γ, α) = π sin α log γ sin α + (1 γ ) + 4γ sin α 1 γ ( )} γ cos α + cos α arcsin. (0.1) 1 + γ In particular, C 1, (γ, α) = γ[π(1 γ )] 1/. For p =, inequality (0.9) and the formula (0.1) for C 1, (γ, α) imply the well-known estimates for Rf(z) Rf(0), If(z) If(0) and f(z) f(0) by Rf Rf(0) (see, for example, [3, 4, 9, 15]). In particular, we obtain the estimate Rf(z) Rf(0) 4 π arctan ( r R generally known as the Schwarz Arcustangens Formula. ) Rf Rf(0), (0.13) 3

4 In general, (0.4) i (0.10) lead to the inequality C 1, p (γ, α) C 0, p (γ, α) which becomes equality for some values of p and α. In particular, this is the case for p =. We also show that C 1, p (γ, π/) = C 0, p (γ, π/), that is the inequality If(z) If(0) S p (r/r) Rf Rf(0) p (0.14) holds with the sharp constant S p (r/r) defined by (0.8). In conclusion we note that constant (0.8) can be written in the form S p (γ) = κ(γ) [ 1 κ (γ) ] 1/( p) πr 1/p B n=0 ( p 1 p, n + 1 ) (p 1)/p κ (γ)} n, (0.15) where B(u, v) is the Beta-function. Inequality (0.14) with the sharp constant (0.15) is a generalization of the classical estimate If(z) If(0) ( ) R + r π log Rf Rf(0), R r due to Carathéodory and Plemelj (see [4, 3]). The present paper extends results of our article [11] dedicated to sharp two-sided parameter dependent estimates for Re iα (f(z) f(0))} by its maximal and minimal values on a circle as well as to the sharp constant for Re iα (f(z) f(0))} involving the value Rf. 1 Estimate of Re iα(z) (f(z) f(0))} by Rf p with associated orthogonality condition We set for real valued functions g and h defined on the circle ζ = R, (g, h) = g(ζ)h(ζ) dζ and we denote the L p -norm, 1 p, of g by g p. We use the notations f(z) = f(z) f(0) and ( ) e iα(z) z G z,α(z) (ζ) = R. (1.1) ζ z Since z plays the role of a parameter in what follows, we frequently do not mark the dependence of α on z. The following assertion is the main objective of this section. Lemma 1. Let f be analytic on D R with continuous real part on D R, 1 p, and let α(z) be a real valued function, z < R. Further, let Rf(ζ)Φ(ζ) dζ = 0, (1.) 4

5 where Φ is a real continuous function on ζ = R. Then for any fixed point z, z = r < R, there holds with the sharp constant Re iα(z) f(z)} C Φ, p (z, α(z)) Rf p (1.3) C Φ, p (z, α) = 1 πr min λ R G z,α λφ q, (1.4) where q = p/(p 1) for 1 < p <, q = for p = 1, and q = 1 for p =. In particular, for any fixed z, z = r < R, there holds f(z) C Φ, p (z, arg f(z)) Rf p. (1.5) Proof. Let Φ C( D R ) and g L q ( D R ) be fixed. Consider the functional F g (h) = g(ζ)h(ζ) dζ, (1.6) on the linear manifold C Φ = h C( D R ) : (h, Φ) = 0} of the space L p ( D R ). We show that F g (h) sup h C Φ h p It follows from the Hahn-Banach theorem that (see [10]) F g (ϕ) sup ϕ L p,ψ ϕ p = min λ R g λφ q. (1.7) = min λ R g λψ q, (1.8) where Ψ L q ( D R ), L p,ψ = ϕ L p ( D R ) : (ϕ, Ψ) = 0}. Suppose the Ψ is continuous. Let 1 p <. Given any ε > 0, for every ϕ L p,ψ there exists ϕ ε C( D R ) such that ϕ ϕ ε p ε. In the case p =, there exists ϕ ε C( D R ) such that the Lebesgue measure of the set E = ζ D R : ϕ(ζ) ϕ ε (ζ)} does not exceed ε and ϕ ε ϕ. Assuming Ψ 0 we write ϕ ε = c ε Ψ + (ϕ ε c ε Ψ), where c ε = (ϕ ε, Ψ)/ Ψ. Then ϕ ε = ϕ ε c ε Ψ C Ψ. First, consider the case 1 p <. If Ψ = 0, then (1.7) follows from (1.8) because C( D R ) is dense in L p ( D R ). For Ψ 0 inequality ϕ ϕ ε p ε implies the estimate ϕ ϕ ε p ε + c ε Ψ p with c ε = (ϕ ε, Ψ) Ψ = (ϕ ε ϕ, Ψ) Ψ ϕ ϕ ε p Ψ q Ψ Ψ q Ψ ε. Hence, ϕ ϕ ε p kε, where k = 1 + Ψ p Ψ q Ψ. Thus C Φ is dense in L p,φ, which, in view of (1.8), implies (1.7). 5

6 Now, let p =. If Ψ = 0, it follows from (1.6) and ϕ ε ϕ that F g (ϕ ε ) ϕ ε F g(ϕ) ϕ g L1 (E). (1.9) Since mes E ε, we have g L1 (E) 0 as ε 0 which leads to (1.7) by (1.8) and (1.9). If Ψ 0, then taking into account the estimates and we obtain F g ( ϕ ε ) F g (ϕ) F g ( ϕ ε ϕ ε ) + F g (ϕ ε ϕ), c ε = (ϕ ε, Ψ) Ψ ϕ ε ϕ ε c ε Ψ, = (ϕ ε ϕ, Ψ) Ψ ϕ Ψ Ψ F g ( ϕ ε ) F g (ϕ) ( k g 1 ε + g L1 (E)) ϕ, where k = Ψ Ψ. This, together with the estimate ϕ ε ϕ + c ε Ψ, implies F g ( ϕ ε ) 1 Fg (ϕ) ( ) } k g 1 ε + g L1 (E). ϕ ε 1 + kε ϕ Combining this with(1.8) and using the arbitrariness of ε we arrive at (1.7) with p =. Let us apply the duality relation (1.7). The Cauchy-Schwarz formula f(z) = i If(0) + 1 ζ + z Rf(ζ)dζ (1.10) πi (ζ z)ζ (see, for example, [3, 1]) represents an analytic function in D R with the real part Rf continuous on D R. Clearly, (1.10) implies f(z) = 1 πr z Rf(ζ) dζ. ζ z (1.11) Using (1.11) and (1.1), we obtain Re iα f(z)} = 1 } πr R e iα z ζ z Rf(ζ) dζ = 1 G z,α (ζ) Rf(ζ) dζ. (1.1) πr Hence and by (1.), the sharp constant C Φ, p (z, α) in can be written in the form Re iα f(z)} C Φ, p (z, α) Rf p (1.13) C Φ, p (z, α) = 1 πr sup 1 h C Φ h p ε, G z,α (ζ)h(ζ) dζ. (1.14) 6

7 Therefore, applying (1.7) to the functional (1.6) with g(ζ) = G z,α (ζ), we arrive at the representation (1.4) for the sharp constant in (1.3). We have proved that inequality (1.3) with the sharp constant C Φ, p (z, α(z)) is valid at any point z D R for any fixed real valued function α(z) on D R and any analytic function f on D R with real part continuous in D R. Having f fixed in (1.3) and choosing α to satisfy α(z) = arg f(z), we arrive at (1.5). and Remark 1. Obviously, (1.4) implies C Φ, (z, α) = 1 πr C 0, (z, α) = 1 πr G z,α, (1.15) ( ) Gz,α (G z,α, Φ) Φ 1/, (1.16) if Φ 0. Let us evaluate G z,α. With the notation ζ = Re it, z = re iτ, γ = r/r one has e iα z ζ z = eiα re iτ Re it re iτ = Setting ϕ = t τ and using (1.17) we obtain [ ( )] e G z,α iα z π+τ = R dζ = γ ζ z +τ π [ ( e = Rγ iα R e iϕ γ and making elementary calculations, we arrive at π which together with (1.18) gives Hence and by (1.15), (1.16) we conclude γe iα e i(t τ) γ. (1.17) [ ( R e iα e i(t τ) γ )] Rdt )] π dϕ = r (cos(ϕ α) γ cos α) dϕ, (1.18) R (1 γ cos ϕ + γ ) (cos(ϕ α) γ cos α) (1 γ cos ϕ + γ ) dϕ = π 1 γ, G z,α = C 0, (z, α) = πr R R r. (1.19) r πr(r r ), (1.0) and provided Φ 0. C Φ, (z, α) = 1 r πr R r (G } z,α, Φ) 1/, (1.1) πr Φ We need one more auxiliary assertion. Its proof, given in [11], is reproduced here for readers convenience. 7

8 Lemma. Let z = r < R. The relations hold min G z,α (ζ) = r(r cos α R) R r, max Proof. Setting ϕ = t τ in (1.17), we obtain ( ) ( ) e iα z γe iα G z,α (ζ) = R = R = ζ z e iϕ γ Consider the function We have g(ϕ) = G z,α (ζ) = r(r cos α + R) R r. (1.) γ(cos(ϕ α) γ cos α) 1 γ cos ϕ + γ. (1.3) cos(ϕ α) γ cos α 1 γ cos ϕ + γ, ϕ π. (1.4) g (ϕ) = (γ 1) cos α sin ϕ + (γ + 1) sin α cos ϕ γ sin α (1 γ cos ϕ + γ ). Solving the equation g (ϕ) = 0, we find and sin ϕ + = (1 γ ) sin α 1 + γ cos α + γ, cos ϕ + = γ + (1 + γ ) cos α 1 + γ cos α + γ, (1.5) sin ϕ = (1 γ ) sin α 1 γ cos α + γ, cos ϕ = γ (1 + γ ) cos α 1 γ cos α + γ, (1.6) where ϕ + and ϕ are critical points of g(ϕ). Setting (1.5) and (1.6) into (1.4) we arrive at g(ϕ + ) = γ cos α + 1, g(ϕ 1 γ ) = γ cos α 1. 1 γ It follows from (1.4) that Since g(ϕ + ) > g(ϕ ) and g() = g(π) = cos α 1 + γ g() = g(π) = g() = g(π) = it follows from (1.3), (1.4) that ( ) e iα z max R ζ z ( ) e iα min R z ζ z which implies (1.). The proof is complete. = γ cos α cos α 1 γ. γ cos α cos α 1 γ γ cos α γ = g(ϕ + ), γ cos α cos α 1 γ γ cos α 1 1 γ = g(ϕ ), = γg(ϕ + ) = γ γ cos α γ = = γg(ϕ ) = γ γ cos α 1 1 γ = r(r cos α + R) R r, r(r cos α R) R r, 8

9 Estimate for Re iα(z) f(z)} by Rf ω p if the mean value of Φ on the circle ζ = R is equal to zero Let us assume that Φ(ζ) dζ = 0. Replacing f with f ω, where ω is a real constant, in Lemma 1, we obtain an estimate for Re iα(z) f(z)}. Proposition 1. Let f be analytic on D R with continuous real part on D R, 1 p, and let α(z) be a real valued function, z < R. Further, let Rf(ζ)Φ(ζ) dζ = 0, (.1) where Φ is a real continuous function on ζ = R, for which Φ(ζ) dζ = 0. (.) Then for any fixed point z, z = r < R, and arbitrary real constant ω there holds Re iα(z) f(z)} C Φ, p (z, α(z)) Rf ω p (.3) with the sharp constant C Φ; p (z, α(z)) given by (1.4). In particular, for any fixed point z, z = r < R, and arbitrary real constant ω, the inequality holds f(z) C Φ, p (z, arg f(z)) Rf ω p. (.4) Remark. The norm Rf ω p in (.3) and (.4) can be replaced with the best approximation E p (Rf) of Rf by a real constant in the norm of the space L p ( D R ) and Note that E p (Rf) = min ω R Rf ω p. (.5) E (Rf) = Rf Rf(0), (.6) E (Rf) = 1 Ω f(r), (.7) where Ω f (R) = A f (R) B f (R) is the oscillation of Rf on the circle ζ = R. Here and in what follow, A f (R) = maxrf(ζ) : ζ = R} and B f (R) = minrf(ζ) : ζ = R}. Indeed, we have 1/ Rf ω = [Rf(ζ) ω] dζ } = π 1/ R [Rf(Re iϕ ) ω] dϕ}, (.8) 9

10 which implies the representation where E (Rf) = min Rf ω = π } [ ] 1/ R Rf(Re iϕ ) A 0 dϕ, ω R which proves (.6). Since the minimum value in A 0 = 1 π Rf(Re iϕ )dϕ = Rf(0) π E (Rf) = min ω R Rf ω is attained at ω = [A f (R) + B f (R)]/ and hence, E (Rf) = Rf A f(r) + B f (R) = A f (R) A f(r) + B f (R) relation (.7) follows. We introduce the set W (α 1, α ) = w C : α 1 arg w α π α arg w π α 1 }, (.9) where 0 α 1 < α π/. The next assertion contains explicit consequences of Proposition 1 for Φ = 0. It also provides an estimate of f(z) for a class of analytic functions in D R with continuous real part in D R for which f(z) W (α 1, α ), z D R. Similarly to Remark, the norm Rf ω p in the next inequalities can be replaced by E p (Rf). Corollary 1. Let f be analytic on D R with continuous real part on D R, 1 p. Further, let α(z) be a real valued function, z < R. Then for any fixed point z, z = r < R, and for an arbitrary real constant ω the inequality holds with the sharp constant C 0, p (z, α(z)), where and Re iα(z) f(z)} C 0, p (z, α(z)) Rf ω p (.10) C 0, p (z, α) = 1 ( r ) R C 1/p 0, p R, α, (.11) C 0, p (γ, α) = γ π cos(ϕ α) γ cos α π 1 γ cos ϕ + γ p/(p 1) dϕ} (p 1)/p (.1) 10

11 if 1 < p <, and C 0, 1 (γ, α) = γ(1 + γ cos α ), (.13) π(1 γ ) } C 0, (γ, α) = 4 sin α log γ sin α + (1 γ cos α) 1/ π (1 γ ) 1/ + cos α arcsin (γ cos α). (.14) In particular, C 0, (γ, α) = γ π(1 γ ). (.15) with If f(z) W (α 1, α ) for z D R, then f(z) max C 0, p (z, α) Rf ω p (.16) α 1 α α max C 0, 1 (z, α) = C 0, 1 (z, α 1 ), (.17) α 1 α α max C 0, (z, α) = C 0, (z, α ). (.18) α 1 α α Proof. We put Φ(z) = 0 in Proposition 1. For p = 1, formula (.11) with the factor (.13) follows directly from (1.4) and (1.). For p =, representation (.11) with the factor (.14) was derived in [11]. Now, suppose 1 < p <. Combining (1.4) with (1.1) and (1.17) we have C 0, p (z, α) = 1 π+τ πr +τ ( ) γe iα p/(p 1) (p 1)/p Rdt} R, e i(t τ) γ which after the change of variable ϕ = t τ becomes C 0, p (z, α) = 1 π ( ) γe iα p/(p 1) (p 1)/p πr 1/p dϕ} R. (.19) e iϕ γ Using the notation C 0, p (γ, α) = 1 π ( ) γe iα p/(p 1) (p 1)/p π dϕ} R, (.0) e iϕ γ we rewrite (.19) as C 0, p (z, α) = 1 ( r ) R C 1/p 0, p R, α, (.1) 11

12 which together with (1.3) proves (.11) and (.1) for 1 < p <. Formula (.11) with p = and the factor (.15) has been already derived (see (1.0)). Now, we pass to the proof of (.16)-(.18). First, we show the equality C 0, p (z, α) = C 0, p (z, α). For p = 1 and p = it follows directly from (.13) and (.14). Suppose 1 < p <. By (.0), C 0, p (γ, α) = γ π ( ) e iα p/(p 1) (p 1)/p π dϕ} R. e iϕ γ Replacing here ϕ by ψ we obtain C 0, p (γ, α) = γ π ( ) e iα p/(p 1) (p 1)/p π dψ} R = C e iψ 0, p (γ, α). γ This, together with (.1), leads to C 0, p (z, α) = C 0, p (z, α). Hence, by (.4) f(z) C 0, p (z, arg f(z)) Rf ω p. (.) Let 0 α π/. By (.0) and (.1), C 0, p (z, π α) = C 0, p (z, α). Combining this with C 0, p (z, α) = C 0, p (z, α) we obtain supc 0, p (z, arg f(z)) : f(z) W (α 1, α )} = maxc 0, p (z, α) : α 1 α α }, which together (.) implies (.16). Equality (.17) follows from the last relation, (.13) and the monotonicity of cos α on [0, π/]. Now, we prove (.18). In view of (.11) and (.14), } C 0, (z, α) = 4 sin α log γ sin α + (1 γ cos α) 1/ π (1 γ ) 1/ + cos α arcsin (γ cos α), where γ = r/r. We study the function C 0, (z, α) for 0 α π/. We have C 0, (z, α) = 4 (cos α log γ sin α + (1 ) γ cos α) 1/ sin α arcsin(γ cos α). (.3) α π (1 γ ) 1/ Using the equalities cos α log γ sin α + (1 γ cos α) 1/ (1 γ ) 1/ = cos α γ sin α 0 dt 1 γ + t, and the estimates γ cos α dt sin α arcsin(γ cos α) = sin α, 0 1 t γ sin α cos α 0 dt 1 γ + t > γ sin α cos α 1 γ + γ sin α, 1

13 γ cos α sin α 0 dt 1 t < γ sin α cos α 1 γ cos α = γ sin α cos α 1 γ + γ sin α, which follow from the mean value theorem for α (0, π/), we obtain from (.3) Thus, C 0, (z, α) increases on [0, π/]. C 0, (z, α) α > 0. Remark 3. Let α = π/. Since the integrand in (.1) is an even function, we have C 0, p (γ, π/) = γ π for the best constant in the inequality π ( 0 sin ϕ 1 γ cos ϕ + γ ) q } 1/q dϕ (.4) I f(z) R 1/p C 0, p (r/r, π/) Rf ω p, (.5) where q = p/(p 1), 1 < p <. Making the change of variable t = cos ϕ and setting κ(γ) = (γ)/(1 + γ ), we find C 0, p (γ, π/) = κ(γ) π which together with (.5) leads to (0.7) and (0.8). The integral m=0 I q (κ) = 1 1 } (1 t ) (q 1)/ 1/q 1 [1 κ(γ)t] dt (.6) q 1 (1 t ) (q 1)/ (1 κt) q dt is the sum of each of two series ( ) (q 1)/ 1 ( 1) m m and ( ) q/ 1 ( 1) m m m=0 1 1 t m (1 κt) q dt t m dt, (1 κt) q (1 t ) 1/ the first of which turns into a finite sum for odd q and the second one for even q. For odd q, the recurrence relation I n+1 (κ) = (n )!! (n 1)!! 1 κ (1 κ ) 1 n κ I n 1(κ) implies I n+1 (κ) = κ n+ n k=1 ( 1) n+k (k )!! (k 1)!! 13 ( ) κ k + ( 1)n 1 + κ log 1 κ κn+1 1 κ.

14 Hence, putting κ = (γ)/(1 + γ ) and taking into account the equality I 1 (κ) = 1 κ log 1 + κ 1 κ and (.6), we find C 0, n+1 (γ, π/) = 1 n π 4( 1) n log 1 + γ 1 γ + (1 + γ ) γ n k=1 ( 1) n+k (k )!! (k 1)!! ( ) } 1 k n+1 γ. 1 γ For example, C 0, 3/ (γ, π/) = 1 π For even q, by the recurrence relation γ(1 + γ ) (1 γ ) 1 log 1 + γ } 1/3. 1 γ I n+ (κ) = π(n 1)!! (n)!! 1 κ (1 κ ) 1 (n+1)/ κ I n(κ), we have I n+ (κ) = Hence, using π κ n+3 n k=1 ( 1) n+k (k 1)!! (k)!! ( ) κ (k+1)/ ( ) + π( 1)n 1 1 κ. 1 κ κ n+ 1 κ I (κ) = π ( 1 1 κ ) κ 1 κ, with κ = (γ)/(1 + γ ) as well as (.6), we obtain C 0, n+ (γ, π/) = 1 n+1 π In particular, 4( 1) n πγ 1 γ + π(1 + γ ) γ n k=1 ( 1) n+k (k 1)!! (k)!! } π(3 γ 1/4 ). ( ) } 1 k+1 n+ γ. 1 γ C 0, 4/3 (γ, π/) = γ π 4(1 γ ) 3 Remark 4. Let α = 0. Since the integrand in (.1) is even, it follows that C 0, p (γ, 0) = γ π π 0 cos ϕ γ 1 γ cos ϕ + γ q dϕ} 1/q, 14

15 where q = p/(p 1), 1 < p <. The change of variable t = cos ϕ implies the equality for the sharp constant in C 0, p (γ, 0) = κ(γ) π 1 t γ q } 1/q dt (.7) 1 [1 κ(γ)t] q (1 t ) 1/ R f(z) C 0, p (r/r, 0)R 1/p Rf ω p. (.8) The integral in (.7) can be evaluated for q = n, that is for p = (n)/(n 1). We introduce the notation By the binomial formula, J n (κ) = 1 κ n J n (κ) = n k=0 1 1 (t γ) n dt. (1 κt) n (1 t ) 1/ ( 1) k (n)!(1 κγ) k k!(n k)! 1 Evaluating the last integral and using (.7), we conclude C 0, n (γ, 0) = (π) 1 n n n n k 1 k=1 m=0 1 ( 1) k+m (n)! ( ) 1 m k 1 k!(n k)!m!(k m 1)! Remark 5. The well known inequalities (see, for instance, [8]) R f(z) 4 π arcsin ( r R ) Rf, dt (1 κt) k (1 t ) 1/. ( ) } 1 m k+1 n 1 + γ. 1 γ I f(z) π log R + r R r Rf are particular cases of inequalities in Proposition 1. They follow from (.10) and (.11) with p =, ω = 0, α = 0, and α = π/, combined with (.14). The inequality f(z) π log R + r R r Rf (see [11]) follows from (.18), where p =, ω = 0, α = π/ and (.14), (.16). The class of inequalities we are studying in this section embraces the following three sharp estimates R f(z) r R r max R f(ζ), (.9) I f(z) Rr R r max R f(ζ), (.30) 15

16 f(z) r R r max R f(ζ) (.31) (see [3, 14, 15] and the bibliography in [3]). Sometimes, (.31) is called Hadamard-Borel-Carathéodory inequality (see, e.g., [3]). For the first time, the inequality f(z) Cr R r max Rf(ζ) (.3) was obtained by Hadamard (real part theorem) with C = 4 in 189 [7] and used in the theory of entire functions. Here f is an analytic function on the disc D R, continuous on D R and vanishing at z = 0. Different proofs of (.3) with C = are given in [1,, 5, 13, 16, 17]. Inequalities(.9)-(.31) follow from Corollary 1 with p = 1, ω = A f (R), that is they are particular cases of the estimate Re iα f(z)} r(r + r cos α ) R r max R f(ζ) for α = 0, α = π/, and α = arg f(z), respectively. Corollary 1 implies one more inequality. Putting p = 1 and ω = A f (R) in (.16) and taking into account (.11), (.13), and (.17), we arrive at the estimate f(z) r(r + r cos α 1 ) R r max R f(ζ), (.33) valid for functions f such that f(z) W (α 1, α ) with z D R. In particular, setting here α 1 = 0 we arrive at (.31). Concluding this section, we make an observation concerning Proposition 1 with p =. Remark 6. Let m be a positive integer and let P m } be the sequence of functions on the circle ζ = R defined by P n (ζ) = R n c k ζ k, (.34) where c 1, c,..., c n C, 1 n m. Let f be an analytic function on D R with continuous real part on D R and let α(z) be a real valued function, z < R. Suppose that Rf(ζ) P n (ζ) dζ = 0 (.35) for all P n P m }. We show that for any fixed z with z = r < R, there holds inequality k=1 Re iα(z) f(z)} 1 ( r ) R K 1/ m E (Rf) (.36) R 16

17 with the sharp constant K m (γ) = γ m+1 π(1 γ ), (.37) where E (Rf) = R f is the best approximation of Rf by a constant in the norm of L ( D R ). Introducing the notation ξ k = Rc k, η k = Ic k, ζ = Re it we write (.34) as the trigonometric polynomial n } P n (ζ) = P n n (Re it ) = R (ξ k + iη k )R k e ikt = (a k cos kt + b k sin kt), (.38) k=1 with a k = ξ k R k, b k = η k R k. Let K m (z, α) denote the sharp constant in k=1 Re iα(z) f(z)} K m (z, α) Rf ω, (.39) where ω is an arbitrary real constant. Taking into account that the mean value of function (.38) on the circle ζ = R is zero, we obtain from Proposition 1 K m (z, α) = 1 πr = 1 πr min min G z,α λ P n = 1 P n P m} λ R πr π min R P n P m} min G z,α P n P n P m} [ G z,α (Re it ) P n (Re it )] dt } 1/. (.40) Let z = re iτ and, as above ζ = Re it, γ = r/r. We have ( ) e iα zζ 1 ( ) } k } z G z,α (ζ) = R = R e iα = R γ k e i[α+k(τ t)] 1 zζ 1 ζ = γ k cos(kt α kτ) = k=1 where β k = α + kτ. Hence and by (.40) K m (z, α) = 1 π R π [ G z,α (Re it ) k=1 k=1 γ k (cos kt cos β k + sin kt sin β k ), (.41) k=1 m γ k (cos kt cos β k + sin kt sin β k )] dt k=1 By (.41), (.4) and the Parseval equality, K m (z, α) = 1 π π R k=m+1 ( ) } 1/ ( γ k cos β k + γ k sin 1 β k = πr 17 k=m+1 1/ γ k ) 1/. (.4),

18 that is K m (z, α) = γ m+1 πr(1 γ ). Using (.37) we find the representation of the sharp constant K m (z, α) = R 1/ K m (r/r) in (.39) which by Remark implies (.36) with the sharp constant (.37). 3 Estimate for Re iα(z) f(z)} by Rf (Rf, Φ)/(1, Φ) p if the mean value of Φ is not zero Suppose Φ(ζ) dζ 0 and replace f with f ω in Lemma 1, where ω is a real constant. By (1.) Rf(ζ) ω}φ(ζ) dζ = 0, that is ω = (Rf, Φ)/(1, Φ). Hence, by Lemma 1 we arrive at another type of estimates for Re iα(z) f(z)}. Proposition. Let f be analytic on D R with continuous real part on D R, 1 p, and let α(z) be a real valued function, z < R. Further, let Φ be a real continuous function on ζ = R, for which the inequality Φ(ζ) dζ 0 (3.1) holds. Then for any fixed point z, z = r < R, there holds Re iα(z) f(z)} C Φ, p (z, α(z)) Rf (Rf, Φ)/(1, Φ) p (3.) with the sharp constant C Φ, p (z, α(z)) given by (1.4). In particular, for any fixed point z, z = r < R, the inequality is valid f(z) C Φ, p (z, arg f(z)) Rf (Rf, Φ)/(1, Φ) p. (3.3) The following assertion is a particular case of Proposition for Φ = 1. Corollary. Let f be analytic on D R with continuous real part on D R, 1 p. Further, let α(z) be a real valued function, z < R. Then for any fixed point z, z = r < R, there holds Re iα(z) f(z)} C 1, p (z, α(z)) R f p (3.4) with the sharp constant C p (z, α(z)), where C 1, p (z, α) = 1 ( r ) R C 1/p 1, p R, α, (3.5) 18

19 and C 1, p (γ, α) = γ π min λ R π cos(ϕ α) γ cos α 1 γ cos ϕ + γ λ p/(p 1) dϕ} (p 1)/p (3.6) if 1 < p <, and C 1, 1 (γ, α) = γ π(1 γ ), (3.7) C 1, (γ, α) = π sin α log γ sin α + (1 γ ) + 4γ sin α 1 γ ( γ cos α + cos α arcsin 1 + γ )}. (3.8) In particular, C 1, (γ, α) = γ π(1 γ ). (3.9) Proof. We set Φ(z) 1 in Proposition. Then (3.) takes the form (3.4). The equality (3.5) with C 1, p (γ, α) = 1 π min λ R π ( ) γe iα R λ e iϕ γ p/(p 1) dϕ} (p 1)/p, (3.10) where 1 < p <, can be derived from (1.4) in the same way as (.1) with the constant (.0) was obtained in Corollary 1. Formula (3.6) follows directly from (3.10) and (1.3). 1. The case p = 1. By (1.4), C 1, 1 (z, α) = 1 πr min max λ R G z,α (ζ) λ. (3.11) Since λ is subject to one of the three alternatives λ min G z,α(ζ), min G z,α(ζ) < λ < max G z,α(ζ), λ max G z,α(ζ), it follows that the minimum with respect to λ in (3.11) is attained at λ = 1 } min G z,α(ζ) + max G z,α(ζ), which by Lemma implies λ = r cos α R r. 19

20 Putting the value of λ into (3.11) and using (1.) we obtain C 1, 1 (z, α) = 1 πr which proves (3.5) with p = 1 and the factor (3.7).. The case p =. From G z,α (z) dζ = R rr R r, } } e iα z ζ z dζ Re iα z = R i(ζ z)ζ dζ = 0, and (1.1) with Φ(ζ) 1 we see that (3.5) holds with p = and the factor (3.9). 3. The case p =. Let the function α and z with z = r < R be fixed. It is well known (see, for example, [10]), that λ gives the minimum in (3.10) with p = if and only if π ( ) } e iα sign R λ dϕ = 0. (3.1) e iϕ γ We show that this equality holds for λ = γ(1 + γ ) 1 cos α with γ [0, 1). We rewrite the left-hand side of the equation ( ) e iα R + γ cos α = 0 (3.13) e iϕ γ 1 + γ as ( ) e iα R e iϕ γ ( ) (1 + γe iϕ )e iα = R (e iϕ γ)(1 + γ ) + γ ( e iα 1 + γ cos α = R We introduce the angle ϑ by the equalities where ) e iϕ γ + γeiα 1 + γ = γ (1 γ ) cos ϕ cos α + (1 + γ ) sin ϕ sin α 1 γ cos ϕ + γ. (3.14) cos ϑ = (1 γ ) k(α, γ) cos α, sin ϑ = (1 + γ ) sin α, (3.15) k(α, γ) k(α, γ) = [(1 γ ) cos α + (1 + γ ) sin α] 1/ = [(1 + γ ) 4γ cos α] 1/. (3.16) From (3.14)-(3.15) we obtain ( ) e iα R e iϕ γ + γ k(α, γ) cos α = 1 + γ 1 + γ cos(ϕ ϑ) 1 γ cos ϕ + γ. (3.17) Thus, the equation (3.13) with unknown ϕ is reduced to cos(ϕ ϑ) = 0. solution of system (3.15) in (, π]. Let ϑ be the 0

21 The distance between two successive roots ϕ n = ϑ π/ + πn, n = 0, ±1, ±,..., of the equation cos(ϕ ϑ) = 0 is equal to π. We put ζ 0 = e iϕ 0, ζ 1 = e iϕ 1 with ϕ 0 = ϑ π/, ϕ 1 = ϑ + π/. Then ( ) e iα R + γ ( ) e iα ζ 0 γ 1 + γ cos α = R + γ cos α = 0. ζ 1 γ 1 + γ Thus, for fixed γ [0, 1) and α, the points ζ 0 and ζ 1 divide the circle ζ = 1 into two half-circles such that on one of them the left-hand side of (3.13) is positive and on another is negative. Hence (3.1) holds with λ = γ(1 + γ ) 1 cos α and, therefore, by (3.5) and (3.10), C 1, (z, α) = C 1, ( r R, α ), (3.18) where This and (3.17) imply C 1, (γ, α) = γ π C 1, (γ, α) = π ( ) e iα R e ϕ γ + γ 1 + γ cos α dϕ. γk(α, γ) π cos(ϕ ϑ) dϕ, (3.19) π(1 + γ ) 1 γ cos ϕ + γ where k(α, γ) is defined by (3.16) and ϑ is the solution of (3.15) in (, π]. Equality (3.19) can be written as γk(α, γ) ϑ+π/ } cos(ϕ ϑ) ϑ+3π/ C 1, (γ, α) = π(1 + γ ) ϑ/ 1 γ cos ϕ + γ dϕ cos(ϕ ϑ) ϑ+π/ 1 γ cos ϕ + γ dϕ. In the first integral we make the change of variable ψ = ϕ and in the second integral we put η = π ϕ. Then γk(α, γ) π/ ϑ } cos(ψ + ϑ) π/ ϑ C 1, (γ, α) = π(1 + γ ) / ϑ 1 γ cos ψ + γ dψ + cos(η + ϑ) / ϑ 1 + γ cos η + γ dη, which implies that is C 1, (γ, α) = C 1, (γ, α) = γk(α, γ) π γk(α, γ) π π/ ϑ / ϑ π/ ϑ / ϑ cos(ψ + ϑ) (1 + γ ) 4γ cos ψ dψ, Substituting the integrals π/ ϑ cos ψ (1 + γ ) 4γ cos ψ dψ = 1 γ(1 γ ) arctan / ϑ 1 cos ψ cos ϑ sin ψ sin ϑ dψ. (3.0) (1 + γ ) 4γ cos ψ ( γ cos ϑ 1 γ ),

22 π/ ϑ / ϑ sin ψ (1 + γ ) 4γ cos ψ dψ = 1 γ(1 + γ ) log 1 + γ + γ sin ϑ 1 + γ γ sin ϑ into (3.0) we obtain C 1, (γ, α) = cos ϑ π k(α, γ) γ cos ϑ arctan 1 γ 1 γ + sin ϑ (1 + γ ) log 1 + } γ + sin ϑ. 1 + γ γ sin ϑ Taking into account (3.15), (3.16), as well as the identity arctan[x(1 x ) 1/ ] = arcsin x, we rewrite the last representation as C 1, (γ, α) = ( ) γ cos α cos α arcsin π 1 + γ + sin α log [(1 } γ ) + 4γ sin α] 1/ + γ sin α. (3.1) 1 γ By (3.1) and (3.18) we arrive at (3.5) with p = with the right-hand side given by (3.8). Remark 7. Comparing the formulas (.13), (3.7) and (.14), (3.8) we conclude that in general C 0, p (γ, α) C 1, p (γ, α). However, for certain values of p and α the equality may hold. This is, clearly, the case for p = in view of (.15) and (3.9). Let us show now that C 0, p (γ, π/) = C 1, p (γ, π/), 1 p, i.e. that sharp constants in I f(z) C 0, p (γ, π/) Rf ω p, (3.) I f(z) C 1, p (γ, π/) R f p, (3.3) coincide. In view of (.11), (3.5), it suffices to prove that C 0, p (γ, π/) = C 1, p (γ, π/). The equalities γ C 0, 1 (γ, π/) = C 1, 1 (γ, π/) = π(1 γ ), C 0, (γ, π/) = C 1, (γ, π/) = π log 1 + γ 1 γ follow directly from (.13), (3.7) and (.14), (3.8). For 1 < p <, by (3.6) C 1, p (γ, π/) = γ π π min λ R sin ϕ 1 γ cos ϕ + γ λ q dϕ} 1/q. (3.4) It is well-known (see, for instance, [10]), that λ gives the minimum in (3.4) if and only if π q 1 ( ) sin ϕ 1 γ cos ϕ + γ λ sin ϕ sign 1 γ cos ϕ + γ λ dϕ = 0.

23 Clearly, the equality holds for λ = 0. Putting λ = 0 in (3.4) and using (.1), we conclude that C 1, p (γ, π/) = C 0, p (γ, π/) for 1 < p <. Thus, Remark 3 relating C 0, p (γ, π/) is also valid for C 1, p (γ, π/) and inequality (0.14) holds with the sharp constant (0.8). We shall write the sharp constant (0.8) in (0.7) and (0.14) in a different form. Using the equality (see, for example, [6]) π 0 ( sin ϕ ) q dϕ = B 1 γ cos ϕ + γ ( q + 1, 1 ) ( F q, q ; q + ) ; γ, where F (a, b; c; x) is the hypergeometric Gauss function, and the relation ( a + 1 F (a, b; a b + 1; x) = (1 x) 1 b (1 + x) b a 1 F b, a ) 4x + 1 b; a b + 1;, (1 + x) we conclude by (.4) that C 0, p (γ, π/) = κ(γ) π κ(γ) [ 1 κ (γ) ] 1/( p) π [ 1 κ (γ) ] ( (1 q)/ q + 1 B, 1 ) ( 1 F, 1; q + 1 )} 1/q ; κ (γ) = B n=0 ( p 1 p, n + 1 ) (p 1)/p κ (γ)} n, (3.5) where κ(γ) = (γ)/(1+γ ). Combining (3.5) with (3.3), (3.5) and the equality C 1, p (γ, π/) = C 0, p (γ, π/) we arrive at (0.15). The next assertion contains an estimate of f(z) for a class of analytic functions in D R with the real part continuous in D R and such that f(z) W (α 1, α ), z D R, where W (α 1, α ) is defined by (.9). Corollary 3. Let f be analytic on D R with continuous real part on D R, 1 p, and let f(z) W (α 1, α ) for z D R. Then the inequality holds f(z) where C 1, p (z, α) is given by (3.5) (3.8). In particular, with C 1, (z, α) defined by (3.5), (3.8). Proof. Putting Φ(ζ) 1 in (3.3) we obtain max C 1, p (z, α) R f p, (3.6) α 1 α α max C 1, (z, α) = C 1, (z, α ), (3.7) α 1 α α f(z) C 1, p (z, arg f(z)) R f p. (3.8) We show that C 1, p (z, α) = C 1, p (z, α). For p = 1 and p =, this follows directly from (3.5), (3.7), and (3.8). 3

24 Let 1 < p <. By (3.10), C 1, p (γ, α) = γ π min λ R π ( ) e iα R λ e iϕ γ which after the change of variable ϕ = π ψ becomes C 1, p (γ, α) = γ π min λ R 0 π ( ) e iα R λ e iψ γ 0 p/(p 1) p/(p 1) dϕ} (p 1)/p, dψ} (p 1)/p = C 1, p (γ, α). This together with (3.5) implies C 1, p (z, α) = C 1, p (z, α). Hence, (3.8) can be written as f(z) C 1, p (z, arg f(z)) R f p. (3.9) Let 0 α π/. By (3.5) and (3.10) we have C 1, p (z, π α) = C 1, p (z, α). This and C 1, p (z, α) = C 1, p (z, α) imply supc 1, p (z, arg f(z)) : f(z) W (α 1, α )} = maxc 1, p (z, α) : α 1 α α }, which together with (3.9) leads to (3.6). Now, we prove (3.7). Owing (3.5) and (3.8), C 1, (z, α) = sin α log γ sin α + ( ) } (1 γ ) + 4γ sin α γ cos α + cos α arcsin, π 1 γ 1 + γ where γ = r/r. Let us consider C 1, (z, α) for 0 α π/. We have C 1, (z, α) = cos α log γ sin α + (1 γ ) + 4γ sin α α π 1 γ Note that the relations cos α log γ sin α + (1 γ ) + 4γ sin α 1 γ = cos α ( ) γ cos α sin α arcsin 1 + γ and the mean value theorem imply γ sin α cos α 0 dt (1 γ ) + t > ( γ cos α sin α arcsin 1 + γ γ sin α γ cos α(1+γ ) 1 = sin α dt (1 γ ) + t, dt 1 t, γ cos α sin α [(1 γ ) + 4γ sin α] 1/, )}. (3.30)

25 γ cos α(1+γ ) 1 sin α 0 dt 1 t < where α (0, π/). Therefore, it follows from (3.30) that γ cos α sin α [(1 γ ) + 4γ sin α] 1/, C 1, (z, α) α > 0. Thus, C 1, (z, α) increases on the interval [0, π/]. Remark 8. The class of inequalities considered in this section include the following three inequalities R f(z) 4 π arctan ( r R ) R f, (3.31) I f(z) π log R + r R r R f, (3.3) f(z) π log R + r R r R f (3.33) (see [3, 4, 9, 15] and the bibliography in [3, 9]). Inequalities (3.31), (3.3) follow from (3.4), (3.5) with p = combined with (3.8) with α = 0 and α = π/, respectively. Inequality (3.33) follows from Corollary 3. In fact, by (3.8), C 1, (γ, 0) = ( ) γ π arcsin = 4 arctan γ, 1 + γ π which together with Corollary 3 leads to C 1, (γ, π/) = π log 1 + γ 1 γ, max C 1, (γ, α) = C 1, (γ, π/) = 0 α π/ π log 1 + γ 1 γ. Acknowledgements. The research of the first author was supported by the KAMEA program of the Ministry of Absorption, State of Israel, and by Research Authority of the College of Judea and Samaria, Ariel. References [1] E. Borel, Démonstration élémentaire d un théorème de M. Picard sur les fonctions entières, C.R. Acad. Sci., 1 (1896),

26 [] E. Borel, Méthodes et problèmes de Théorie des Fonctions, Gauthier-Villars, Paris, 19. [3] R.B. Burckel, An introduction to classical complex analysis, V. 1, Academic Press, New York - San Francisco, [4] C. Carathéodory, Elementarer Beweis für den Fundamentalsatz der konformen Abbildungen, Mathematische Abhandlungen H.A. Schwarz Gewidmet, 19-41, Verlag Julius Springer (1914), Berlin; Reprinted by Chelsea Publishing Co. (1974), New York. [5] M.L. Cartwright, Integral Functions, Oxford Univ. Press, Oxford, [6] I.S. Gradshtein and I.M. Ryzhik; Alan Jeffrey, editor, Table of integrals, series and products, Fifth edition, Academic Press, New York, [7] J. Hadamard, Sur les fonctions entières de la forme e G(X), C.R. Acad. Sci., 114 (189), [8] A. Hurwitz and R. Courant, Funktionentheorie, Vierte Auflage, Springer-Verlag, Berlin - Göttingen - Heidelberg - New-York, [9] P. Koebe, Über das Schwarzsche Lemma und einige damit zusammenhängende Ungleicheitsbeziehungen der Potentialtheorie und Funktionentheorie, Math. Zeit., 6 (190), [10] N. Korneichuk, Exact constants in approximation theory. Encyclopedia of Mathematics and its applications, 38, Cambridge University Press, [11] G.I. Kresin and V.G. Maz ya, Sharp parametric inequalities for analytic and harmonic functions related to Hadamard-Borel-Carathéodory inequalities, Functional Differential Equations, 9, N. 1- (00), [1] B.Ya. Levin, Lectures on Entire Functions, Transl. of Math. Monogfaphs, v.150, Amer. Math. Soc., Providence, [13] J.E. Littlewood, Lectures on the theory of functions, Oxford Univ. Press, Oxford, [14] D.S. Mitrinović, Analytic Inequalities, Springer-Verlag, Berlin-Heidelberg-New York, [15] G. Polya and G. Szegö, Problems and Theorems in Analysis, v. 1, Springer-Verlag, Berlin - Heidelberg - New-York, 197. [16] E.G. Titchmarsh, The theory of functions, Cambrige University Press, New York, 196. [17] L. Zalcman, Picard s theorem without tears, Amer. Math. Monthly, 85: 4 (1978),

Uniform Convergence of Fourier Series Michael Taylor

Uniform Convergence of Fourier Series Michael Taylor Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula

Διαβάστε περισσότερα

2 Composition. Invertible Mappings

2 Composition. Invertible Mappings Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

Section 8.3 Trigonometric Equations

Section 8.3 Trigonometric Equations 99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

Example Sheet 3 Solutions

Example Sheet 3 Solutions Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note

Διαβάστε περισσότερα

Matrices and Determinants

Matrices and Determinants Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z

Διαβάστε περισσότερα

Every set of first-order formulas is equivalent to an independent set

Every set of first-order formulas is equivalent to an independent set Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

Homework 3 Solutions

Homework 3 Solutions Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β

3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β 3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018

SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals

Διαβάστε περισσότερα

Partial Differential Equations in Biology The boundary element method. March 26, 2013

Partial Differential Equations in Biology The boundary element method. March 26, 2013 The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet

Διαβάστε περισσότερα

Areas and Lengths in Polar Coordinates

Areas and Lengths in Polar Coordinates Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ.

Problem Set 9 Solutions. θ + 1. θ 2 + cotθ ( ) sinθ e iφ is an eigenfunction of the ˆ L 2 operator. / θ 2. φ 2. sin 2 θ φ 2. ( ) = e iφ. = e iφ cosθ. Chemistry 362 Dr Jean M Standard Problem Set 9 Solutions The ˆ L 2 operator is defined as Verify that the angular wavefunction Y θ,φ) Also verify that the eigenvalue is given by 2! 2 & L ˆ 2! 2 2 θ 2 +

Διαβάστε περισσότερα

ST5224: Advanced Statistical Theory II

ST5224: Advanced Statistical Theory II ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known

Διαβάστε περισσότερα

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:

HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch: HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying

Διαβάστε περισσότερα

C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics

Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)

Διαβάστε περισσότερα

Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for

Διαβάστε περισσότερα

PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)

PROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1) GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk

Διαβάστε περισσότερα

Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We

Διαβάστε περισσότερα

Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential. be continuous functions on the interval Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

Concrete Mathematics Exercises from 30 September 2016

Concrete Mathematics Exercises from 30 September 2016 Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)

Διαβάστε περισσότερα

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ. Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

Congruence Classes of Invertible Matrices of Order 3 over F 2

Congruence Classes of Invertible Matrices of Order 3 over F 2 International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and

Διαβάστε περισσότερα

Section 7.6 Double and Half Angle Formulas

Section 7.6 Double and Half Angle Formulas 09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)

Διαβάστε περισσότερα

CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

EE512: Error Control Coding

EE512: Error Control Coding EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

4.6 Autoregressive Moving Average Model ARMA(1,1)

4.6 Autoregressive Moving Average Model ARMA(1,1) 84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

6.1. Dirac Equation. Hamiltonian. Dirac Eq.

6.1. Dirac Equation. Hamiltonian. Dirac Eq. 6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2

Διαβάστε περισσότερα

Solutions to Exercise Sheet 5

Solutions to Exercise Sheet 5 Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X

Διαβάστε περισσότερα

Homomorphism in Intuitionistic Fuzzy Automata

Homomorphism in Intuitionistic Fuzzy Automata International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic

Διαβάστε περισσότερα

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3

Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all

Διαβάστε περισσότερα

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p) Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

Math221: HW# 1 solutions

Math221: HW# 1 solutions Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin

Διαβάστε περισσότερα

Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1

Arithmetical applications of lagrangian interpolation. Tanguy Rivoal. Institut Fourier CNRS and Université de Grenoble 1 Arithmetical applications of lagrangian interpolation Tanguy Rivoal Institut Fourier CNRS and Université de Grenoble Conference Diophantine and Analytic Problems in Number Theory, The 00th anniversary

Διαβάστε περισσότερα

PARTIAL NOTES for 6.1 Trigonometric Identities

PARTIAL NOTES for 6.1 Trigonometric Identities PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot

Διαβάστε περισσότερα

D Alembert s Solution to the Wave Equation

D Alembert s Solution to the Wave Equation D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique

Διαβάστε περισσότερα

6.3 Forecasting ARMA processes

6.3 Forecasting ARMA processes 122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

A Note on Intuitionistic Fuzzy. Equivalence Relation

A Note on Intuitionistic Fuzzy. Equivalence Relation International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com

Διαβάστε περισσότερα

A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering

A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix

Διαβάστε περισσότερα

Reminders: linear functions

Reminders: linear functions Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U

Διαβάστε περισσότερα

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- ----------------- Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

n=2 In the present paper, we introduce and investigate the following two more generalized

n=2 In the present paper, we introduce and investigate the following two more generalized MATEMATIQKI VESNIK 59 (007), 65 73 UDK 517.54 originalni nauqni rad research paper SOME SUBCLASSES OF CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS Zhi-Gang Wang Abstract. In the present paper, the author

Διαβάστε περισσότερα

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1

Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1 Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the

Διαβάστε περισσότερα

Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type

Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Noname manuscript No. will be inserted by the editor Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Victor Nijimbere Received: date / Accepted: date Abstract

Διαβάστε περισσότερα

SPECIAL FUNCTIONS and POLYNOMIALS

SPECIAL FUNCTIONS and POLYNOMIALS SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195

Διαβάστε περισσότερα

The k-α-exponential Function

The k-α-exponential Function Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,

Διαβάστε περισσότερα

The Simply Typed Lambda Calculus

The Simply Typed Lambda Calculus Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and

Διαβάστε περισσότερα

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions

SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)

Διαβάστε περισσότερα

Solution Series 9. i=1 x i and i=1 x i.

Solution Series 9. i=1 x i and i=1 x i. Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x

Διαβάστε περισσότερα

Second Order Partial Differential Equations

Second Order Partial Differential Equations Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

Lecture 15 - Root System Axiomatics

Lecture 15 - Root System Axiomatics Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the

Διαβάστε περισσότερα

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω 0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +

Διαβάστε περισσότερα

The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients

The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients The Number of Zeros of a Polynomial in a Disk as a Consequence of Restrictions on the Coefficients Robert Gardner Brett Shields Department of Mathematics and Statistics Department of Mathematics and Statistics

Διαβάστε περισσότερα

Other Test Constructions: Likelihood Ratio & Bayes Tests

Other Test Constructions: Likelihood Ratio & Bayes Tests Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :

Διαβάστε περισσότερα

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?

ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =? Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least

Διαβάστε περισσότερα

( y) Partial Differential Equations

( y) Partial Differential Equations Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate

Διαβάστε περισσότερα

w o = R 1 p. (1) R = p =. = 1

w o = R 1 p. (1) R = p =. = 1 Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:

Διαβάστε περισσότερα

Tridiagonal matrices. Gérard MEURANT. October, 2008

Tridiagonal matrices. Gérard MEURANT. October, 2008 Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,

Διαβάστε περισσότερα

SOME PROPERTIES OF FUZZY REAL NUMBERS

SOME PROPERTIES OF FUZZY REAL NUMBERS Sahand Communications in Mathematical Analysis (SCMA) Vol. 3 No. 1 (2016), 21-27 http://scma.maragheh.ac.ir SOME PROPERTIES OF FUZZY REAL NUMBERS BAYAZ DARABY 1 AND JAVAD JAFARI 2 Abstract. In the mathematical

Διαβάστε περισσότερα

Lecture 13 - Root Space Decomposition II

Lecture 13 - Root Space Decomposition II Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).

Διαβάστε περισσότερα

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R + Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα

Second Order RLC Filters

Second Order RLC Filters ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor

Διαβάστε περισσότερα

Commutative Monoids in Intuitionistic Fuzzy Sets

Commutative Monoids in Intuitionistic Fuzzy Sets Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,

Διαβάστε περισσότερα

Integrals in cylindrical, spherical coordinates (Sect. 15.7)

Integrals in cylindrical, spherical coordinates (Sect. 15.7) Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.

Διαβάστε περισσότερα

Finite Field Problems: Solutions

Finite Field Problems: Solutions Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The

Διαβάστε περισσότερα

Problem Set 3: Solutions

Problem Set 3: Solutions CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C

Διαβάστε περισσότερα

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1. Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

Διαβάστε περισσότερα

ExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation

ExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied

Διαβάστε περισσότερα

Quadratic Expressions

Quadratic Expressions Quadratic Expressions. The standard form of a quadratic equation is ax + bx + c = 0 where a, b, c R and a 0. The roots of ax + bx + c = 0 are b ± b a 4ac. 3. For the equation ax +bx+c = 0, sum of the roots

Διαβάστε περισσότερα

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in

Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that

Διαβάστε περισσότερα

Fractional Colorings and Zykov Products of graphs

Fractional Colorings and Zykov Products of graphs Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is

Διαβάστε περισσότερα

Spherical Coordinates

Spherical Coordinates Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical

Διαβάστε περισσότερα

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.

Διαβάστε περισσότερα

Lecture 2. Soundness and completeness of propositional logic

Lecture 2. Soundness and completeness of propositional logic Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness

Διαβάστε περισσότερα

Statistical Inference I Locally most powerful tests

Statistical Inference I Locally most powerful tests Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided

Διαβάστε περισσότερα

New bounds for spherical two-distance sets and equiangular lines

New bounds for spherical two-distance sets and equiangular lines New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a

Διαβάστε περισσότερα

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)

Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required) Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts

Διαβάστε περισσότερα

Forced Pendulum Numerical approach

Forced Pendulum Numerical approach Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.

Διαβάστε περισσότερα

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.

Διαβάστε περισσότερα

Trigonometric Formula Sheet

Trigonometric Formula Sheet Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ

Διαβάστε περισσότερα

DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation

DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values

Διαβάστε περισσότερα

Logsine integrals. Notes by G.J.O. Jameson. log sin θ dθ = π log 2,

Logsine integrals. Notes by G.J.O. Jameson. log sin θ dθ = π log 2, Logsine integrals Notes by G.J.O. Jameson The basic logsine integrals are: log sin θ dθ = log( sin θ) dθ = log cos θ dθ = π log, () log( cos θ) dθ =. () The equivalence of () and () is obvious. To prove

Διαβάστε περισσότερα

If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2

If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013 Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering

Διαβάστε περισσότερα

The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions

The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions Pure Mathematical Sciences, Vol. 1, 01, no. 4, 187-196 The Fekete Szegö Theorem for a Subclass of Quasi-Convex Functions Goh Jiun Shyan School of Science and Technology Universiti Malaysia Sabah Jalan

Διαβάστε περισσότερα

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM

SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max

Διαβάστε περισσότερα

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed

Διαβάστε περισσότερα

If we restrict the domain of y = sin x to [ π 2, π 2

If we restrict the domain of y = sin x to [ π 2, π 2 Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points

The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points Applied Mathematical Sciences, Vol. 3, 009, no., 6-66 The Negative Neumann Eigenvalues of Second Order Differential Equation with Two Turning Points A. Neamaty and E. A. Sazgar Department of Mathematics,

Διαβάστε περισσότερα

Srednicki Chapter 55

Srednicki Chapter 55 Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third

Διαβάστε περισσότερα

Homework 8 Model Solution Section

Homework 8 Model Solution Section MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

A General Note on δ-quasi Monotone and Increasing Sequence

A General Note on δ-quasi Monotone and Increasing Sequence International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in

Διαβάστε περισσότερα

Parametrized Surfaces

Parametrized Surfaces Parametrized Surfaces Recall from our unit on vector-valued functions at the beginning of the semester that an R 3 -valued function c(t) in one parameter is a mapping of the form c : I R 3 where I is some

Διαβάστε περισσότερα

The challenges of non-stable predicates

The challenges of non-stable predicates The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates

Διαβάστε περισσότερα