# COMPUTATION OF THE MITTAG-LEFFLER FUNCTION E α,β (z) AND ITS DERIVATIVE. Abstract

Save this PDF as:

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

Download "COMPUTATION OF THE MITTAG-LEFFLER FUNCTION E α,β (z) AND ITS DERIVATIVE. Abstract"

## Transcript

1 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION E α,β (z) AND ITS DERIVATIVE Rudolf Gorenflo 1, JouliaLoutchko 1 & Yuri Luchko 2 Dedicated to Francesco Mainardi, Professor at the University of Bologna, on the occasion of his 60-th birthday Abstract In this paper algorithms for numerical evaluation of the Mittag-Leffler function E α,β (z) = k=0 z k, α > 0, β R, z C Γ(β + αk) and its derivative for all values of the parameters α > 0, β R and all values of the argument z C are presented. For different parts of the complex plane different numerical techniques are used. In every case we provide estimates for accuracy of the computation; numerous pictures showing the behaviour of the Mittag- Leffler function for different values of the parameters and on different lines in the complex plane are included. The ideas und techniques employed in the paper can be used for numerical evaluation of other functions of the hypergeometric type. In particular, the same method with some small modifications can be applied for the Wright function which plays a very important role in the theory of partial differential equations of fractional order. Mathematics Subject Classification: 33E12, 65D20, 33F05, 30E15 Key Words and Phrases: Mittag-Leffler function, integral representations of special functions, numerical computation of special functions, Mathematica programs

2 2 R. Gorenflo, J. Loutchko and Yu. Luchko The function E α (z) = 1. Introduction k=0 z k,α>0, z C, (1) Γ(1 + αk) was introduced by Mittag-Leffler in 1902 in connection with his method of summing divergent series. It is suited to serve as a very simple example of an entire function of a specified order 1/α and of normal type. Further investigations of properties of this function and its applications to various questions of mathematical analysis have been carried out by Wiman [33]-[34] and Buhl [7]. An important generalization, E α,β (z) = k=0 z k, α > 0, β C, z C, (2) Γ(β + αk) was introduced by Humbert [22], Agarwal [1], Humbert and Agarwal [23], Cleota and Hughes [9]. In recent years the interest in functions of Mittag-Leffler type among scientists, engineers and applications-oriented mathematicians has deepened. This interest is caused by the close connection of these functions to differential equations of fractional (meaning: non-integer) order and integral equations of Abel type, such equations becoming more and more popular in modelling natural and technical processes and situations (Babenko [2], Bagley and Torvik [3], Blair [4]-[5], Caputo and Mainardi [8], El-Sayed [12], Gorenflo et al. [16], Gorenflo and Mainardi [17]- [19], Gorenflo and Rutman [20], Gorenflo and Vessella [21], Luchko [24], Luchko and Srivastava [27], Mainardi [28], Schneider and Wyss [30], Slonimski [31], Westerlund and Ekstam [32]). Although there already exists a rich literature on analytical methods for solving differential equations of fractional order, it is to be remarked that solutions in closed form have been found only for such equations with constant coefficients and for a rather small class of equations with particular variable coefficients. In general, numerical solution techniques are required. In connection with the basic role of Mittag-Leffler type functions for the solution of fractional differential equations and integral equations of Abel type it seems important as a first step to develop their theory and stable methods for their numerical computation. The most simple function of Mittag-Leffler type, E α (z), depends on two variables: the complex argument z and the real parameter α. The generalizations need at least one more argument, and α may be allowed to be complex. Experience in the computation of special functions of mathematical physics teach us that in distinct parts of the complex plane different numerical techniques should be used. The aim of this paper is the development of some

3 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION... 3 methods for computing the Mittag-Leffler function (2) and its derivative as well as the assessment of the range of their applicability and accuracy. We propose Taylor series in the case 0 <α 1 for small z, asymptotic representations for z of large magnitude, and special integral representations for intermediate values of the argument z. By aid of a recursion formula we reduce the case 1 <αto the case 0 <α 1. We devise the needed algorithms in the system MATHEMATICA. 2. Integral representations of the Mittag-Leffler function Integral representations play a prominent role in the analysis of entire functions. For the Mittag-Leffler function (2) such representations in form of an improper integral along the Hankel loop have been treated in the case β =1and in the general case with arbitrary β by Erdélyi et al. [13] and Dzherbashyan [10], [11]. They considered E α,β (z) = 1 e ζ1/α ζ (1 β)/α dζ, z G ( ) (ɛ; δ), (3) 2πiα γ(ɛ;δ) ζ z E α,β (z) = 1 α z(1 β)/α e z1/α + 1 e ζ1/α ζ (1 β)/α dζ, z G (+) (ɛ; δ), (4) 2πiα γ(ɛ;δ) ζ z under the conditions 0 <α<2, πα 2 <δ min{π, πα}. (5) The contour γ(ɛ; δ) can be seen in Figure 1. It consists of two rays S δ and S δ (arg ζ = δ, ζ ɛ and arg ζ = δ, ζ ɛ) and a circular arc C δ (0; ɛ) ( ζ = ɛ, δ arg ζ δ). On its left side there is a region G ( ) (ɛ, δ), on its right side a region G (+) (ɛ; δ). Using the integral representations in (3), (4) it is not difficult to get asymptotic expansions for the Mittag-Leffler function in the complex plane. Let 0 <α<2, β be an arbitrary number, and δ be chosen to satisfy the condition (5). Then we have, for any p N and z : Whenever arg z δ, E α,β (z) = z (1 β) α e z1/α α Whenever δ arg z π, E α,β (z) = p k=1 p k=1 z k Γ(β αk) + O( z 1 p ). (6) z k Γ(β αk) + O( z 1 p ). (7)

4 4 R. Gorenflo, J. Loutchko and Yu. Luchko These formulas are also used in our numerical algorithm. In what follows we restrict our attention to the case β R, the most important in the applications. For the purpose of numerical computation we look for integral representations better suited than (3) und (4). With eζ1/α ζ (1 β)/α φ(ζ,z)= ζ z we then have I = 1 φ(ζ,z) dζ = 1 φ(ζ,z) dζ (8) 2πiα γ(ɛ;δ) 2πiα S δ + 1 φ(ζ,z) dζ + 1 φ(ζ,z) dζ = I 1 + I 2 + I 3. 2πiα C δ (0;ɛ) 2πiα S δ Now we transform the integrals I 1,I 2 and I 3.ForI 1 we take ζ = re iδ,ɛ r< and get I 1 = 1 φ(ζ,z) dζ = 1 ɛ e (re iδ ) 1/α (re iδ ) (1 β)/α 2πiα S δ 2πiα + re iδ e iδ dr. (9) z Analogously for (with ζ = re iδ,ɛ r< ) I 3 = 1 φ(ζ,z) dζ = 1 + e (reiδ ) 1/α (re iδ ) (1 β)/α 2πiα S δ 2πiα ɛ re iδ e iδ dr. (10) z S δ C δ (0; ɛ) G (+) (ɛ; δ) G ( ) (ɛ; δ) δ ɛ γ(ɛ; δ) S δ Figure 1: The contour γ(ɛ; δ)

5 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION... 5 For I 2 we have ζ = ɛe iϕ, δ ϕ δ and with I 2 = 1 φ(ζ,z) dζ = 1 δ e (ɛeiϕ ) 1/α (ɛe iϕ ) (1 β)/α 2πiα C δ (0;ɛ) 2πiα δ ɛe iϕ ɛie iϕ dϕ (11) z = ɛ1+(1 β)/α 2πα δ δ e ɛ1/α e iϕ/α e iϕ((1 β)/α+1) ɛe iϕ dϕ = z P [α, β, ɛ, ϕ, z] = ɛ1+(1 β)/α 2πα We rewrite the sum I 1 + I 3 as with δ δ P [α, β, ɛ, ϕ, z] dϕ, e ɛ1/α cos(ϕ/α) (cos(ω)+isin(ω)) ɛe iϕ, (12) z ω = ɛ 1/α sin(ϕ/α)+ϕ(1 + (1 β)/α). I 1 + I 3 = 1 2πiα + ɛ { e (re iδ ) 1/α (re iδ ) (1 β)/α re iδ z e(re iδ ) 1/α (re iδ ) (1 β)/α re iδ e iδ} dr z = 1 + { r (1 β)/α e r1/α cos(δ/α) e i(r1/α sin(δ/α)+δ(1+(1 β)/α)) 2πiα ɛ re iδ z e i(r1/α sin(δ/α)+δ(1+(1 β)/α)) } + re iδ dr = K[α, β, δ, r, z] dr, z K[α, β, δ, r, z] = 1 πα r(1 β)/α e r1/α cos(δ/α) r sin(ψ δ) z sin(ψ) r 2 2rz cos(δ)+z 2, (13) ψ = r 1/α sin(δ/α)+δ(1 + (1 β)/α). By aid of (8)-(13) we can rewrite the formulas (3) and (4) as + δ E α,β (z) = K[α, β, δ, r, z] dr + P [α, β, ɛ, ϕ, z] dϕ, z G ( ) (ɛ; δ), (14) ɛ δ + δ E α,β (z) = K[α, β, δ, r, z] dr + P [α, β, ɛ, ϕ, z] dϕ (15) ɛ δ + 1 α z(1 β)/α e z1/α, z G (+) (ɛ; δ). Let us now consider the case 0 <α 1, z 0. By condition (5) we can choose δ =min{π, πα} = πα. Then the kernel function (13) looks simpler: ɛ K[α,β,πα,r,z]= K[α, β, r, z] (16) e iδ

6 6 R. Gorenflo, J. Loutchko and Yu. Luchko = 1 r sin(π(1 β)) z sin(π(1 β + α)) πα r(1 β)/α e r1/α r 2 2rz cos(πα)+z 2. In the formulas (14)-(16) for computation of the function E α,β (z) atthearbitrary point z C, z 0 we will distinguish three possibilities for arg z, namely A) arg z >πα, B) arg z = πα, C) arg z <πα. First we conside the case A): arg z >πα. G ( ) (ɛ; πα) G (+) (ɛ; πα) z πα ɛ γ(ɛ; πα) Figure 2: The case arg z >πα In this case z always (for arbitrary ɛ) lies in the region G ( ) (ɛ; δ) (seefigure 2) and we arrive at Theorem 2.1. Under the conditions 0 <α 1, β R, arg z >πα, z 0 the function E α,β (z) has the representations E α,β (z) = ɛ πα K[α, β, r, z] dr + P [α, β, ɛ, ϕ, z] dϕ, ɛ > 0, β R, (17) πα E α,β (z) = 0 K[α, β, r, z] dr, if β<1+α, (18)

7 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION... 7 E α,β (z) = sin(πα) πα with 0 e r1/α r 2 2rz cos(πα)+z 2 dr 1, if β =1+α (19) z K[α, β, r, z] = 1 πα r(1 β)/α e r1/α r sin(π(1 β)) z sin(π(1 β + α)) r 2 2rz cos(πα)+z 2, P [α, β, ɛ, ϕ, z] = ɛ1+(1 β)/α 2πα e ɛ1/α cos(ϕ/α) (cos(ω)+isin(ω)) ɛe iϕ, z ω = ɛ 1/α sin(ϕ/α)+ϕ(1 + (1 β)/α). P r o o f. The representation (17) follows immediately from (14) with δ = πα and (16) and holds for arbitrary β R. Let us now consider (17) for β<1+α. In this case we have P [α, β, ɛ, ϕ, z] Cɛ 1+(1 β)/α, πα ϕ πα, 0 <ɛ<1 with a constant C not depending on ϕ. Consequently, πα P [α, β, ɛ, ϕ, z] dϕ 0 if ɛ 0. (20) Further we have πα This means: If β < 1 + α the integral K[α, β, r, z] =O(r (1 β)/α ), r 0. 0 K[α, β, r, z] dr is a convergent improper integral with singularity at r = 0, but it is there not singular if β 1. Using (20), we can calculate a limit for ɛ 0 in the representation (17) and so arrive at the representation (18). Finally, in the case β =1+α we have the formulas P [α, β, ɛ, ϕ, z] = 1 e ɛ1/α cos(ϕ/α) (cos(ɛ 1/α sin(ϕ/α)) + i sin(ɛ 1/α sin(ϕ/α))) 2πα ɛe iϕ, z πα πα lim P [α, β, ɛ, ϕ, z] dϕ = lim P [α, β, ɛ, ϕ, z] dϕ (21) ɛ 0 πα πα ɛ 0 πα ( = 1 ) dϕ = 1 πα 2παz z, the corresponding integral converging uniformly with respect to ɛ. Hence for β =1+α there holds the formula K[α, β, r, z] = sin(πα) πα e r1/α r 2 2rz cos(πα)+z 2 (22)

8 8 R. Gorenflo, J. Loutchko and Yu. Luchko and the integral 0 K[α, β, r, z] dr is non-singular in the point r = 0. The representation (19) now follows from (17), (21) und (22). Example: Letβ =1,0<α<1, z = t α < 0. Theorem 2.1 then yields the representation E α,1 ( t α )= 1 e r1/α 0 πα which by insertion of r = x α t α can be transformed to E α,1 ( t α )= 1 0 π e xt t α sin(πα) r 2 +2rt α dr, cos(πα)+t2α x α 1 sin(πα) x 2α +2x α dx. (23) cos(πα)+1 For the representation (23) which can be obtained by the Laplace transform method see Gorenflo and Mainardi [17]. ThenextcaseweconsideristhecaseB): arg z = πα. G ( ) (ɛ; πα) G (+) (ɛ; πα) z πα ɛ γ(ɛ; πα) Figure 3: The case arg z = πα In this case we are not allowed to make ɛ arbitrarily small in our contour γ(ɛ; δ), because z is directly lying on this contour if ɛ< z (see Figure 3).

9 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION... 9 Theorem 2.2. Under the conditions 0 <α 1, β R, arg z = πα, z 0 the function E α,β (z) has the representation E α,β (z) = ɛ πα K[α, β, r, z] dr + P [α, β, ɛ, ϕ, z] dϕ, ɛ > z, (24) πα with K[α, β, r, z] = 1 πα r(1 β)/α e r1/α r sin(π(1 β)) z sin(π(1 β + α)) r 2 2rz cos(πα)+z 2, P [α, β, ɛ, ϕ, z] = ɛ1+(1 β)/α 2πα e ɛ1/α cos(ϕ/α) (cos(ω)+isin(ω)) ɛe iϕ, z ω = ɛ 1/α sin(ϕ/α)+ϕ(1 + (1 β)/α). Proof. Ifɛ> z then z is lying in the region G ( ) (ɛ; δ)andtherepresentation (24) follows from (14) with δ = πα and (16). Finally let us discuss the case C): arg z <πα. G ( ) (ɛ; πα) z G (+) (ɛ; πα) πα ɛ γ(ɛ; πα) Figure 4: The case arg z <πα In this case, if 0 <ɛ< z, thenz is in the region G (+) (ɛ; δ), and we have

10 10 R. Gorenflo, J. Loutchko and Yu. Luchko Theorem 2.3. Under the conditions 0 <α 1, β R, arg z <πα, z 0 the function E α,β (z) possesses the representations E α,β (z) = ɛ K[α, β, r, z] dr + πα πα P [α, β, ɛ, ϕ, z] dϕ (25) E α,β (z) = α z(1 β)/α e z1/α, 0 <ɛ< z, β R, K[α, β, r, z] dr + 1 α z(1 β)/α e z1/α, if β<1+α, (26) E α,β (z) = sin(πα) πα 0 e r1/α r 2 dr (27) 2rz cos(πα)+z2 1 z + 1 αz ez1/α, if β =1+α with K[α, β, r, z] = 1 πα r(1 β)/α e r1/α r sin(π(1 β)) z sin(π(1 β + α)) r 2 2rz cos(πα)+z 2, P [α, β, ɛ, ϕ, z] = ɛ1+(1 β)/α 2πα e ɛ1/α cos(ϕ/α) (cos(ω)+isin(ω)) ɛe iϕ, z ω = ɛ 1/α sin(ϕ/α)+ϕ(1 + (1 β)/α). P r o o f. Similar to the proof of Theorem 2.1; use (15) instead of (14). 3. Computation of the Mittag-Leffler function E α,β (z) We have proved that for arbitrary z 0and0<α 1 the Mittag- Leffler function E α,β (z) can be represented by one of the formulas (17)- (19), (24)-(26). We intend to use these formulas for numerical computation if q< z, 0 <q<1 and 0 <α 1. In the case z q, 0 <q<1 we compute E α,β (z) for arbitrary α>0 by aid of the power series (2). The case 1 <αcan be reduced to the case 0 <α 1 by aid of a recursion formula. For computation of the function E α,β (z) for arbitrary z C with arbitrary indices α>0, β R, we will distinguish three possibilities: A) z q, 0 <q<1(q is a fixed number), 0 <α, B) z >q, 0 <α 1and C) z >q, 1 <α. In each case we compute the Mittag-Leffler function with the prescribed accuracy ρ>0. First we consider the case A): z q, 0 <q<1.

11 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION Theorem 3.1. In the case z q, 0 <q<1, 0 <αthe Mittag-Leffler function can be computed with the prescribed accuracy ρ > 0 by use of the formula E α,β (z) = k 0 k=0 z k + µ(z), µ(z) ρ, (28) Γ(β + αk) k 0 =max{[(1 β)/α]+1, [ln(ρ(1 z ))/ ln( z )]}. P r o o f. We transcribe the definition (2) into the form E α,β (z) = m k=0 z k + µ(z,m), µ(z,m) = Γ(β + αk) For all k k 0 [(1 β)/α] + 1 we have the inequality Γ(β + αk) 1 and thus the estimate (m +1 k 0 ) µ(z,m) = z k Γ(β + αk) k=m+1 k=m+1 k=m+1 z k Γ(β + αk). z k = z m z. The estimate µ(z) := µ(z,k 0 ) <ρfollows from the inequality k 0 [ln(ρ(1 z ))/ ln( z )]. Remark 3.1. For computation of the function E α,β (z) in case A) it is recommendable to choose in Theorem 3.1 a number q not very close to 1 (in order to have not a large number of terms in the sum of formula (28)). In our computer programs we have taken q =0.9. We proceed with the case B): q< z, 0 <α 1. In this case we use the integral representations (17)-(19), (24)-(26). We then must compute numerically either the improper integral I = a K[α, β, r, z] dr, a {0,ɛ}, K[α, β, r, z] = 1 r sin(π(1 β)) z sin(π(1 β + α)) πα r(1 β)/α e r1/α r 2 2rz cos(πα)+z 2, or even the integral J = πα πα P [α, β, ɛ, ϕ, z] = ɛ1+(1 β)/α 2πα P [α, β, ɛ, ϕ, z] dϕ, ɛ > 0, e ɛ1/α cos(ϕ/α) (cos(ω)+isin(ω)) ɛe iϕ, z

12 12 R. Gorenflo, J. Loutchko and Yu. Luchko ω = ɛ 1/α sin(ϕ/α)+ϕ(1 + (1 β)/α). The second integral J (the integrand P [α, β, ɛ, ϕ, z] being bounded and the limits of integration being finite) can be calculated with prescribed accuracy ρ>0by one of many product quadrature methods. For calculating the first (improper) integral I over the bounded function K[α, β, r, z] weuse I = Theorem 3.2. The representation a K[α, β, r, z] dr = is valid under the conditions r0 a K[α, β, r, z] dr + µ(z), µ(z) ρ, a {0,ɛ} (29) 0 <α 1, 0 <q< z, { max{1, 2 z, ( ln(πρ/6)) α }, if β 0, r 0 = max{( β +1) α, 2 z, ( 2ln(πρ/(6( β + 2)(2 β ) β )) α }, if β<0. P r o o f. We try to estimate the absolute value of in the expression µ(z,r) = R K[α, β, r, z] dr I = R a K[α, β, r, z] dr + µ(z,r), a {0,ɛ}. We first consider an estimate of the function K[α, β, r, z]. Let z 0 = e iπα and r 2 z. Wethenhave 1 r 2 2rz cos(πα)+z 2 = 1 r 2 z r z z 0 r z 0 1 r 2 ( z z 0 )( z z 0 ) = 1 r 2 ( 1 z ) 2 4 r 2 r and hence also K[α, β, r, z] 1 r + z πα r(1 β)/α e r1/α r 2 2rz cos(πα)+z 2 1 6r πα r(1 β)/α e r1/α r 2 = 6 πα r(1 β)/α 1 e r1/α. Hencewehave,forR 2 z, the estimate r µ(z,r) R r K[α, β, r, z] dr (30)

13 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION R πα r(1 β)/α 1 e r1/α dr = 6 t β e t dt = 6 π R 1/α π Γ(1 β,r1/α ), with the incomplete gamma function (see Erdélyi et al. [13]) Γ(α, x) = More estimates can be obtained by aid of x e t t α 1 dt, x > 0 Lemma 3.1. For the incomplete gamma function Γ(1 β,x) the following estimates hold: Γ(1 β,x) e x,x 1, β 0, (31) Γ(1 β),x) ( β +2)x β e x, x β +1, β < 0. (32) Proof. Fort x 1andβ 0 we have the inequality t β e t e t and hence also (31) from Γ(1 β,x) x e t dt = e x. If β<0 we determine n N such that (n +1) β< n and treat the integral Γ(1 β,x) by n-fold partial integration: Γ(1 β,x) =x β e x βx β 1 e x + β(β +1)x β 2 e x (33) n ( 1) n (β + j)x β n e x +( 1) n+1 j=0 x n (β + j)t β n 1 e t dt. j=0 Now β + n +1 0andx β +1 1 in the last integral, and we can use the inequality (31): t β n 1 e t dt e x. For x β +1and (n +1) β< n we have x x β > βx β 1 > β(β +1)x β 2 >...> β(β +1)...(β + n). The latter inequalities together with the representation(33) yield the estimate Γ(1 β,x) (n +2)x β e x ( β +2)x β e x, x β +1, and thus Lemma 3.1 is proved. Let us return to the proof of Theorem 3.2. estimate (30) and Lemma 3.1 yield the { 6 µ(z,r) π e R1/α, β 0, R 1 6 π ( β +2)R β/α e R1/α, β < 0, R 1/α β +1. (34)

14 14 R. Gorenflo, J. Loutchko and Yu. Luchko For β 0wehave µ(z) := µ(z,r 0 ) <ρ,ifr 0 =max{1, 2 z, ( ln(πρ/6)) α }, and for β<0 we should find an r 0 such that the inequality is fulfilled. We solve this exercise by aid of 6 ( β +2)r β/α 0 e r1/α 0 ρ (35) π Lemma 3.2. For arbitrary x, y, q > 0 there holds the inequality x y (qy) y e x/q. (36) Proof.Withx = ay, a > 0, the inequality (36) can be rewritten in the form (ay) y (qy) y e ay/q which is equivalent to a y q y e ay/q and hence also to ( a q ) y ( e a/q) y. Thelatterinequalityistruebecauseof Hence Lemma 3.2 is proved. e a/q max{a/q, 1}, a,q > 0. Denoting r 1/α 0 by x, β by y, and taking q = 2 we rewrite the inequality (35) by aid of (36) in the form 6 ( β +2)r β/α 0 e r1/α 0 = 6 π π ( β +2)xy e x 6 π ( β + 2)(2y)y e x/2 e x = 6 π ( β + 2)(2 β ) β e r1/α 0 /2 <ρ. Solving for r 0 we find µ(z) := µ(z,r 0 ) ρ for β<0andr 0 =max{( β + 1) α, 2 z, ( 2ln(πρ/(6( β + 2)(2 β ) β )) α }. ThelastcasewehavetodiscussisthecaseC):q< z, 1 <α. Here we use the recursion formula (see Dzherbashyan [11]) E α,β (z) = 1 m m 1 h=0 E α/m,β (z 1/m e i2πh/m )(m 1). (37) In order to reduce case C) to the cases B) and A) we take m = α +1 in formula (37). Then 0 <α/m 1, and we calculate the functions E α/m,β (z 1/m e 2πih/m )as in case A) if z 1/m q<1, and as in case B) if z 1/m >q.

15 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION Remark 3.2. The ideas und techniques employed for the Mittag-Leffler function can be used for numerical calculation of other functions of the hypergeometric type. In particular, the same method with some small modifications can be applied for the Wright function playing a very important role in the theory of partial differential equations of fractional order (see for example Buckwar and Luchko [6], Gorenflo et al. [15], Luchko [25], Luchko and Gorenflo [26], Mainardi et al. [29]). To this end, the following representations (see Gorenflo et al. [14]) can be used instead of the representations (2), (3), and (4): φ(ρ, β; z) = φ(ρ, β; z) = 1 2πi k=0 Ha z k, ρ > 1, β C, k!γ(ρk + β) e ζ+zζ ρ ζ β dζ, ρ > 1, β C, where Ha denotes the Hankel path in the ζ-plane with a cut along the negative real semi-axis arg ζ = π. 4. Computation of the derivative of the function E α,β (z) In many questions of analysis the derivative of a function plays an important role. The derivative of the Mittag-Leffler function can be used for example in iterative methods for determination of its zeros in the complex plane. The function E α,β (z) being an entire function, we find by term-wise differentiation of its power series (2) the representation E α,β(z) = k=1 kz k 1 Γ(β + αk) = (k +1)z k Γ(α + β + αk). (38) For numerical calculation of the function E α,β (z) with prescribed accuracy ρ we distinguish two cases: A) z q<1(q isafixednumber)andb) z >q. We start with the case A): z q<1. Theorem 4.1. In the case z q<1 the derivative (38) of the Mittag-Leffler function can be calculated by aid of the formula E α,β(z) = k 0 k=0 k=0 (k +1)z k + µ(z), µ(z) ρ, (39) Γ(α + β + αk) k 0 =max{k 1, [ln(ρ(1 z ))/ ln( z )]}, [(2 α β)/(α 1)] + 1, α > 1, k 1 = [(3 [ α β)/α]+1, ] [ 0 <α 1, D 0, ] max{ 3 α β α +1, 1 2ωα+ D 2α +1}, 0 <α 1, D > 0, 2 (40)

16 16 R. Gorenflo, J. Loutchko and Yu. Luchko with prescribed accuracy ρ>0. with ω = α + β 3 2,D= α2 4αβ +6α +1 P r o o f. By formula (38) we have E α,β (z) = m k=0 µ(z,m) = (k +1)z k Γ(α + β + αk) + µ(z,m) k=m+1 (k +1)z k Γ(α + β + αk) and can estimate the absolute value of µ(z,m). Now Γ(x) 1forx 1. So, we have the inequalities if if Γ(α + β + αk) =(α + β + αk 1)Γ(α + β + αk 1) α + β + αk 1 k + 1 (41) α>1, k k 1 =[(2 α β)/(α 1)] = 1, Γ(α + β + αk) =(α + β + αk 1)(α + β + αk 2)Γ(α + β + αk 2) (42) (α + β + αk 1)(α + β + αk 2) k +1 0 <α 1, k k 1, with the natural number k 1 given by (40). From (41), (42) follows the estimate µ(z,m) and we finally get k=m+1 (k +1)z k Γ(α + β + αk) k=m+1 z k = z m+1 1 z, m+1 k 1, µ(z) := µ(z,k 0 ) ρ, k 0 =max{k 1, [ln(ρ(1 z ))/ ln( z )]}. Now we consider the case B): z >q. In this case we use the formula (see Dzherbashyan [11]) E α,β (z) =E α,β 1(z) (β 1)E α,β (z), (43) αz which reduces the calculation of the derivative of the Mittag-Leffler function to that of the function E α,β (z), already worked out.

17 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION Numerical Algorithm The numerical scheme for computation of the Mittag-Leffler function given in pseudocode notation by the algorithm below is based on the results presented in Section 3. The algorithm uses the defining series (2) for arguments z of small magnitude, its asymptotic representations (6), (7) for arguments z of large magnitude, and special integral representations for intermediate values of the argument that include a monotonic part K(α, β, χ, z) dχ and an oscillatory part P (α, β, ɛ, φ, z) dφ, which can be evaluated using standard techniques. GIVEN α>0, β R, z C, ρ>0then IF1 <αthen k 0 = α +1 E α,β (z) = 1 k0 1 k 0 k=0 E α/k 0,β(z 1 k 0 exp( 2πik k 0 )) ELSIF z =0THEN E α,β (z) = 1 Γ(β) ELSIF z < 1THEN k 0 =max{ (1 β) α E α,β (z) = k 0 k=0 ELSIF z > α THEN, ln[ρ(1 z )]/ ln( z ) } z k Γ(β+αk) k 0 = ln(ρ)/ ln( z ) IF arg z < απ min{π, απ} THEN E α,β (z) = 1 α z (1 β) α e z1/α k 0 z k k=1 Γ(β αk) ELSE E α,β (z) = k 0 z k k=1 Γ(β αk) ELSE { max{1, 2 z, ( ln( πρ 6 χ 0 = ))α }, β 0 max{( β +1) α πρ, 2 z, ( 2ln( [6( β +2)(2 β ) β ] ))α }, β < 0 K(α, β, χ, z) = 1 απ χ (1 β) α exp( χ 1 α ) χ sin[π(1 β)] z sin[π(1 β+α)] χ 2 2χz cos(απ)+z 2 P (α, β, ɛ, φ, z) = 1 (1 β) 2απ ɛ1+ α exp(ɛ 1 α cos( φ cos(ω)+i sin(ω) α )) ɛ exp(iφ) z ω = φ(1 + (1 β) α )+ɛ 1 α sin( φ α ) IF arg z >απthen IF β 1THEN E α,β (z) = χ 0 0 K(α, β, χ, z) dχ ELSE E α,β (z) = χ 0 1 K(α, β, χ, z) dχ + απ απ ELSIF arg z <απthen IF β 1THEN E α,β (z) = χ 0 0 K(α, β, χ, z) dχ + 1 α z (1 β) α ELSE P (α, β, 1,φ,z) dφ e z1/α

18 18 R. Gorenflo, J. Loutchko and Yu. Luchko E α,β (z) = χ 0 z K(α, β, χ, z) dχ + απ z απ P (α, β, 2,φ,z) dφ + 1 α z (1 β) α 2 ELSE E α,β (z) = χ 0 ( z +1) K(α, β, χ, z) dχ + απ ( z +1) απ P (α, β, 2,φ,z) dφ 2 END e z1/α Remark 5.1. The formulas for E α,β (z) in this algorithm are in error at most by ρ. It is advisable to take ρ = ε m = machine precision. 6. Figures In this section, some figures generated by aid of the methods described in the paper are presented. We have produced them by using the programming system MATHEMATICA Figure 5: The function E α,β ( t) forα =0.25, β = 1 and its derivative. Black line: E α,β ( t), grey line: d dt (E α,β( t))

19 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION Figure 6: The function E α,β ( t) forα =1.75, β = 1 and its derivative. Black line: E α,β ( t), grey line: d dt (E α,β( t)) Figure 7: The function E α,β ( t) forα =2.25, β = 1 and its derivative. Black line: E α,β ( t), grey line: d dt (E α,β( t))

20 20 R. Gorenflo, J. Loutchko and Yu. Luchko Figure 8: E α,β (z) for α =0.75, β =1, arg(z) = απ Figure 9: E α,β (z) 1 α for α =0.75, β =1, arg(z) = απ 2

21 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION Figure 10: E α,β (z) 0forα =0.75, β =1, arg(z) = 3απ Figure 11: E α,β (z) forα =0.75, β =1, arg(z) =π

22 22 R. Gorenflo, J. Loutchko and Yu. Luchko Figure 12: E α,β (z) for α =1.25, β =1, arg(z) = απ Figure 13: E α,β (z) 1 α for α =1.25, β =1, arg(z) = απ 2

23 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION Figure 14: E α,β (z) 0forα =1.25, β =1, arg(z) = 3απ Figure 15: E α,β (z) forα =1.25, β =1, arg(z) =π

24 24 R. Gorenflo, J. Loutchko and Yu. Luchko References [1] R.P. A g a r w a l, A propos d une note de M. Pierre Humbert, C. R. Acad. Sci. Paris, 236(1953), [2] Yu.I. B a b e n k o, Heat and Mass Transfer, Leningrad, Khimiya, 1986 (in Russian). [3] R.L. B a g l e y and P.J. T o r v i k, On the appearance of the fractional derivative in the behaviour of real materials, J. Appl. Mech. 51(1984), [4] G.W.S. B l a i r, Psychorheology: links between the past and the present, Journal of Texture Studies, 5(1974), [5] G.W.S. B l a i r, The role of psychophysics in rheology, J. of Colloid Sciences, 1947, [6] E. B u c k w a r and Yu. L u c h k o, Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations, Journal of Mathematical Analysis and Applications, 227(1998), [7] A. B u h l, Séries analytiques. Sommabilité, Mem. Sci. Math. Fasc. 7, Gauthier-Villars, Paris. [8] M. C a p u t o and F. M a i n a r d i, Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento, SerII, 1(1971), [9] C l e o t a and H u g h e s, Asymptotic developments of certain integral functions, Duke Math. J., 9(1942), [10] M.M. D z h e r b a s h y a n, About integral representation of functions continuous on some rays (generalization of Fourier integral), Izv. Akad. Nauk Armyan. SSR, Ser. Mat., 18(1954), (In Russian). [11] M.M. D z h e r b a s h y a n, Integral Transforms and Representations of Functions in the Complex Plane, Nauka, Moscow, 1966 (in Russian). [12] A.M.A. E l - S a y e d, Fractional order diffusion-wave equation, J. of Theoretical Physics 35, No 2 (1996), [13] A. E r d élyi,w.magnus,f.oberhettingerandf.g.trico mi,higher Transcendental Functions, Vol.3,NewYork,McGraw-HillBook Co., 1954.

25 COMPUTATION OF THE MITTAG-LEFFLER FUNCTION [14] R. G o r e n f l o, Yu. L u c h k o and F. M a i n a r d i, Analytical properties and applications of the Wright function, Fractional Calculus and Applied Analysis 2(1999), [15]R.Gorenflo,Yu.LuchkoandF.Mainardi,Wrightfunctions as scale-invariant solutions of the diffusion-wave equation, Journal of Computational and Applied Mathematics 118(2000), [16] R. G o r e n f l o, Yu. L u c h k o and P.P. Z a b r e j k o, On solvability of linear fractional differential equations in Banach spaces, Fractional Calculus and Applied Analysis 2(1999), [17] R. G o r e n f l o and F. M a i n a r d i, Fractional oscillations and Mittag- Leffler functions, Preprint No. A-14/96, Free University of Berlin, Berlin, [18] R. G o r e n f l o and F. M a i n a r d i, The Mittag-Leffler function in the Riemann-Liouville fractional calculus, In: Kilbas A.A. (eds.) Boundary value problems, special functions and fractional calculus, Minsk, 1996, [19] R. G o r e n f l o and F. M a i n a r d i, Fractional Calculus: Introduction to Integral and Differential Equations of Fractional Order, in: Fractals and Fractional Calculus in Continuum Mechanics (Ed. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, [20] R. G o r e n f l o and R. R u t m a n, On ultraslow and intermediate processes, In: RusevP.,DimovskiI.andKiryakovaV.(eds.)Transform Methods and Special Functions, Sofia, 1994, [21] R. G o r e n f l o and S. V e s s e l l a, Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics 1461, Springer-Verlag, Berlin, [22] P. H u m b e r t, Quelques resultats relatifs a la fonction de Mittag-Leffler, C. r. Acad. sci. Paris, 236(1953), [23] P. H u m b e r t and R.P. A g a r w a l, Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations, Bull.sci.math.(2), 77(1953), [24] Yu. L u c h k o, Operational method in fractional calculus, Fractional Calculus and Applied Analysis 2(1999), [25] Yu. L u c h k o, Asymptotics of zeros of the Wright function, Journal for Analysis and its Applications 19(2000),

26 26 R. Gorenflo, J. Loutchko and Yu. Luchko [26] Yu. L u c h k o and R. G o r e n f l o, Scale-invariant solutions of a partial differential equation of fractional order, Fractional Calculus and Applied Analysis, 1(1998), [27] Yu. L u c h k o and H.M. S r i v a s t a v a, The exact solution of certain differential equations of fractional order by using operational calculus, Computers Math. Appl. 29(1995), [28] F. M a i n a r d i, Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics, in: Fractals and Fractional Calculus in Continuum Mechanics (Ed. A. Carpinteri and F. Mainardi), Springer Verlag, Wien, [29] F. M a i n a r d i, Yu. L u c h k o and G. P a g n i n i, The fundamental solution of the space-time fractional diffusion equation, Fractional Calculus and Applied Analysis 4(2001), [30] W.R. S c h n e i d e r and W. W y s s, Fractional diffusion and wave equations, J. Math. Phys. 30(1989), [31] G.L. S l o n i m s k i, About a law of deformation of polymers, Dokl. Akad. Nauk SSSR (Physic), 140(1961), No 2, [32] S. W e s t e r l u n d and L. E k s t a m, Capacitor theory, IEEE Trans. on Dielectrics and Electrical Insulation 1(1994), [33] A. W i m a n, Über den Fundamentalsatz in der Theorie der Funktionen E α (x), Acta Math., 29(1905), [34] A. W i m a n, Über die Nullstellen der Funktionen E α (x), Acta Math., 29(1905), ) Department of Mathematics and Computer Science Received: Free University of Berlin Arnimallee 3 D Berlin GERMANY 2 ) Department of Business Informatics European University Frankfurt(Oder) POB 1786 D Frankfurt (Oder) GERMANY

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Strain gauge and rosettes

Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

### Variational Wavefunction for the Helium Atom

Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer

Διαβάστε περισσότερα

### 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

### 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

### Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

### 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

### 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

### Math 6 SL Probability Distributions Practice Test Mark Scheme

Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry

Διαβάστε περισσότερα

### 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

### 9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values

Διαβάστε περισσότερα

### 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

### 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

### 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

### 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

### 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

### ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

Διαβάστε περισσότερα

### Integral representations and asymptotic behaviour of a mittag-leffler type function of two variables

Integral representations and asymptotic behaviour of a mittag-leffler type function of two variables Christian Lavault To cite this version: Christian Lavault. Integral representations and asymptotic behaviour

Διαβάστε περισσότερα

### 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

### CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS

CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3

Διαβάστε περισσότερα

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Finite difference method for 2-D heat equation

Finite difference method for 2-D heat equation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen

Διαβάστε περισσότερα

### Section 9.2 Polar Equations and Graphs

180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

### 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

### 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

### 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

### 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

### 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

### On a four-dimensional hyperbolic manifold with finite volume

BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

### ΚΒΑΝΤΙΚΟΙ ΥΠΟΛΟΓΙΣΤΕΣ

Ανώτατο Εκπαιδευτικό Ίδρυμα Πειραιά Τεχνολογικού Τομέα Τμήμα Ηλεκτρονικών Μηχανικών Τ.Ε. ΚΒΑΝΤΙΚΟΙ ΥΠΟΛΟΓΙΣΤΕΣ Πτυχιακή Εργασία Φοιτητής: ΜIΧΑΗΛ ΖΑΓΟΡΙΑΝΑΚΟΣ ΑΜ: 38133 Επιβλέπων Καθηγητής Καθηγητής Ε.

Διαβάστε περισσότερα

### ( ) 2 and compare to M.

Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

### Exercises to Statistics of Material Fatigue No. 5

Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can

Διαβάστε περισσότερα

### Appendix S1 1. ( z) α βc. dβ β δ β

Appendix S1 1 Proof of Lemma 1. Taking first and second partial derivatives of the expected profit function, as expressed in Eq. (7), with respect to l: Π Π ( z, λ, l) l θ + s ( s + h ) g ( t) dt λ Ω(

Διαβάστε περισσότερα

### 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

### g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King

Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i

Διαβάστε περισσότερα

### Problem 3.1 Vector A starts at point (1, 1, 3) and ends at point (2, 1,0). Find a unit vector in the direction of A. Solution: A = 1+9 = 3.

Problem 3.1 Vector A starts at point (1, 1, 3) and ends at point (, 1,0). Find a unit vector in the direction of A. Solution: A = ˆx( 1)+ŷ( 1 ( 1))+ẑ(0 ( 3)) = ˆx+ẑ3, A = 1+9 = 3.16, â = A A = ˆx+ẑ3 3.16

Διαβάστε περισσότερα

### Bessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!

Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation

Διαβάστε περισσότερα

### ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ

ΕΙΣΑΓΩΓΗ ΣΤΗ ΣΤΑΤΙΣΤΙΚΗ ΑΝΑΛΥΣΗ ΕΛΕΝΑ ΦΛΟΚΑ Επίκουρος Καθηγήτρια Τµήµα Φυσικής, Τοµέας Φυσικής Περιβάλλοντος- Μετεωρολογίας ΓΕΝΙΚΟΙ ΟΡΙΣΜΟΙ Πληθυσµός Σύνολο ατόµων ή αντικειµένων στα οποία αναφέρονται

Διαβάστε περισσότερα

### Instruction Execution Times

1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables

Διαβάστε περισσότερα

### 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

### The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality

The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,

Διαβάστε περισσότερα

### 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

### Space Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines

Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the

Διαβάστε περισσότερα

### 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

### Roman Witu la 1. Let ξ = exp(i2π/5). Then, the following formulas hold true [6]:

Novi Sad J. Math. Vol. 43 No. 1 013 9- δ-fibonacci NUMBERS PART II Roman Witu la 1 Abstract. This is a continuation of paper [6]. We study fundamental properties applications of the so called δ-fibonacci

Διαβάστε περισσότερα

### 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

### HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? What is the 50 th percentile for the cigarette histogram?

HISTOGRAMS AND PERCENTILES What is the 25 th percentile of a histogram? The point on the horizontal axis such that of the area under the histogram lies to the left of that point (and to the right) What

Διαβάστε περισσότερα

### 2. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν.

Experiental Copetition: 14 July 011 Proble Page 1 of. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν. Ένα μικρό σωματίδιο μάζας (μπάλα) βρίσκεται σε σταθερή απόσταση z από το πάνω μέρος ενός

Διαβάστε περισσότερα

### Eigenvalues and eigenfunctions of a non-local boundary value problem of Sturm Liouville differential equation

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 5 (2016, pp. 3885 3893 Research India Publications http://www.ripublication.com/gjpam.htm Eigenvalues and eigenfunctions

Διαβάστε περισσότερα

### Durbin-Levinson recursive method

Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again

Διαβάστε περισσότερα

### Orbital angular momentum and the spherical harmonics

Orbital angular momentum and the spherical harmonics March 8, 03 Orbital angular momentum We compare our result on representations of rotations with our previous experience of angular momentum, defined

Διαβάστε περισσότερα

### SOLVING CUBICS AND QUARTICS BY RADICALS

SOLVING CUBICS AND QUARTICS BY RADICALS The purpose of this handout is to record the classical formulas expressing the roots of degree three and degree four polynomials in terms of radicals. We begin with

Διαβάστε περισσότερα

### Mock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =

Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n

Διαβάστε περισσότερα

### Démographie spatiale/spatial Demography

ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΙΑΣ Démographie spatiale/spatial Demography Session 1: Introduction to spatial demography Basic concepts Michail Agorastakis Department of Planning & Regional Development Άδειες Χρήσης

Διαβάστε περισσότερα

### Bounding Nonsplitting Enumeration Degrees

Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,

Διαβάστε περισσότερα

### Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

Διαβάστε περισσότερα

### 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

### High order interpolation function for surface contact problem

3 016 5 Journal of East China Normal University Natural Science No 3 May 016 : 1000-564101603-0009-1 1 1 1 00444; E- 00030 : Lagrange Lobatto Matlab : ; Lagrange; : O41 : A DOI: 103969/jissn1000-56410160300

Διαβάστε περισσότερα

### Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων. Εξάμηνο 7 ο

Εργαστήριο Ανάπτυξης Εφαρμογών Βάσεων Δεδομένων Εξάμηνο 7 ο Procedures and Functions Stored procedures and functions are named blocks of code that enable you to group and organize a series of SQL and PL/SQL

Διαβάστε περισσότερα

### Partial Trace and Partial Transpose

Partial Trace and Partial Transpose by José Luis Gómez-Muñoz http://homepage.cem.itesm.mx/lgomez/quantum/ jose.luis.gomez@itesm.mx This document is based on suggestions by Anirban Das Introduction This

Διαβάστε περισσότερα

### The Spiral of Theodorus, Numerical Analysis, and Special Functions

Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6

Διαβάστε περισσότερα

### ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006

Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή

Διαβάστε περισσότερα

### MATHEMATICS. 1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81

1. If A and B are square matrices of order 3 such that A = -1, B =3, then 3AB = 1) -9 2) -27 3) -81 4) 81 We know that KA = A If A is n th Order 3AB =3 3 A. B = 27 1 3 = 81 3 2. If A= 2 1 0 0 2 1 then

Διαβάστε περισσότερα

### Μηχανική Μάθηση Hypothesis Testing

ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider

Διαβάστε περισσότερα

### Ηλεκτρονικοί Υπολογιστές IV

ΠΑΝΕΠΙΣΤΗΜΙΟ ΙΩΑΝΝΙΝΩΝ ΑΝΟΙΚΤΑ ΑΚΑΔΗΜΑΪΚΑ ΜΑΘΗΜΑΤΑ Ηλεκτρονικοί Υπολογιστές IV Εισαγωγή στα δυναμικά συστήματα Διδάσκων: Επίκουρος Καθηγητής Αθανάσιος Σταυρακούδης Άδειες Χρήσης Το παρόν εκπαιδευτικό υλικό

Διαβάστε περισσότερα

### «Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων. Η μεταξύ τους σχέση και εξέλιξη.»

ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΑΓΡΟΝΟΜΩΝ ΚΑΙ ΤΟΠΟΓΡΑΦΩΝ ΜΗΧΑΝΙΚΩΝ ΤΟΜΕΑΣ ΓΕΩΓΡΑΦΙΑΣ ΚΑΙ ΠΕΡΙΦΕΡΕΙΑΚΟΥ ΣΧΕΔΙΑΣΜΟΥ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ: «Χρήσεις γης, αξίες γης και κυκλοφοριακές ρυθμίσεις στο Δήμο Χαλκιδέων.

Διαβάστε περισσότερα

### Probability and Random Processes (Part II)

Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation

Διαβάστε περισσότερα

### Testing for Indeterminacy: An Application to U.S. Monetary Policy. Technical Appendix

Testing for Indeterminacy: An Application to U.S. Monetary Policy Technical Appendix Thomas A. Lubik Department of Economics Johns Hopkins University Frank Schorfheide Department of Economics University

Διαβάστε περισσότερα

### The ε-pseudospectrum of a Matrix

The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems

Διαβάστε περισσότερα

### ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ ΒΑΛΕΝΤΙΝΑ ΠΑΠΑΔΟΠΟΥΛΟΥ Α.Μ.: 09/061. Υπεύθυνος Καθηγητής: Σάββας Μακρίδης

Α.Τ.Ε.Ι. ΙΟΝΙΩΝ ΝΗΣΩΝ ΠΑΡΑΡΤΗΜΑ ΑΡΓΟΣΤΟΛΙΟΥ ΤΜΗΜΑ ΔΗΜΟΣΙΩΝ ΣΧΕΣΕΩΝ ΚΑΙ ΕΠΙΚΟΙΝΩΝΙΑΣ ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ «Η διαμόρφωση επικοινωνιακής στρατηγικής (και των τακτικών ενεργειών) για την ενδυνάμωση της εταιρικής

Διαβάστε περισσότερα

### Memoirs on Differential Equations and Mathematical Physics

Memoirs on Differential Equations and Mathematical Physics Volume 31, 2004, 83 97 T. Tadumadze and L. Alkhazishvili FORMULAS OF VARIATION OF SOLUTION FOR NON-LINEAR CONTROLLED DELAY DIFFERENTIAL EQUATIONS

Διαβάστε περισσότερα

### TMA4115 Matematikk 3

TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet

Διαβάστε περισσότερα

### 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

### CORDIC Background (4A)

CORDIC Background (4A Copyright (c 20-202 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later

Διαβάστε περισσότερα

### ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 24/3/2007

Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Όλοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα μικρότεροι του 10000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Αν κάπου κάνετε κάποιες υποθέσεις

Διαβάστε περισσότερα

### ON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS

Pa J Statist 2009 Vol 25(2), 135-140 ON NEGTIVE MOMENTS OF CERTIN DISCRETE DISTRIBUTIONS Masood nwar 1 and Munir hmad 2 1 Department of Maematics, COMSTS Institute of Information Technology, Islamabad,

Διαβάστε περισσότερα

### ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΕΠΑΝΑΣΧΕΔΙΑΣΜΟΣ ΓΡΑΜΜΗΣ ΣΥΝΑΡΜΟΛΟΓΗΣΗΣ ΜΕ ΧΡΗΣΗ ΕΡΓΑΛΕΙΩΝ ΛΙΤΗΣ ΠΑΡΑΓΩΓΗΣ REDESIGNING AN ASSEMBLY LINE WITH LEAN PRODUCTION TOOLS

ΔΙΑΤΜΗΜΑΤΙΚΟ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΣΤΗ ΔΙΟΙΚΗΣΗ ΤΩΝ ΕΠΙΧΕΙΡΗΣΕΩΝ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ ΕΠΑΝΑΣΧΕΔΙΑΣΜΟΣ ΓΡΑΜΜΗΣ ΣΥΝΑΡΜΟΛΟΓΗΣΗΣ ΜΕ ΧΡΗΣΗ ΕΡΓΑΛΕΙΩΝ ΛΙΤΗΣ ΠΑΡΑΓΩΓΗΣ REDESIGNING AN ASSEMBLY LINE WITH

Διαβάστε περισσότερα