University College Cork: MA2008 Complex Numbers and Functions Exercises Prove:
|
|
- Διόδοτος Δουρέντης
- 6 χρόνια πριν
- Προβολές:
Transcript
1 University College Cork: MA8 Complex Numbers and Functions 5 Exercises. Show that (a) i, i i, i, i 5 i,... i i, i, i i,.... Let + i and 5i. Find in Cartesian form: (a) ( + ) (c) (d) (e) Im + i 7 i ( i) (f) Re + i. For x + iy, find in Cartesian form: (a) Im (Im ). Prove: (c) Im(/) (a) Any complex number is equal to the conjugate of its conjugate. is real if and only if. (c) i i (d) Re(i) Im (e) Im(i) Re (f) If then at least one of and must be ero. 5. Find (a) 5i (c) + (d) + i + i (e) (f) (g) ( + i) 6 i ( + i) (h) + i (i) cos θ + i sin θ 6. Represent in modulus argument form: 7. Determine the principal argument of (a) 7 i (c) + i (d) i 8. Show that multiplication by i corresponds to an anti-clockwise rotation about the origin through the angle π/. (a) + i + i (c) 8 (d) i 9. Verify the triangle inequality for i, + i.. Identify each of the following loci, and represent them on the Argand diagram: (a) Re (c) Re( ) (d) arg < π (e) π < Im π (f) + + (g) + (h) + i i (i) < Im <
2 University College Cork: MA8 Complex Numbers and Functions 5 Exercises. Determine all solutions to the following equations, and plot them on the Argand diagram: (a) i i (c) i (d) i (e) 5 (f) 8 (g) + i (h) + i (i) (j) 7 8 (k) 6 (l) i (m) 6 (n) + 8 (o) (p) For x + iy, demonstrate that the two square roots of are ( ) + x x ± + i(sign y) where sign y and the real square roots have positive sign. {, y, y < (Hint: Set w u + iv with w, and separate into two real equations to express u and v in terms of x and y.). Use Question to find the square roots of (a) i 8i (c) 5 + i (d) i. Use Question to find the solutions to the following equations: (a) (5 + i) i ( + i) 8 6i
3 University College Cork: MA8 Complex Numbers and Functions 5. Find f( + i), f( i) and f( + i) for each of the following functions: (a) f() f() Exercises, f() Re otherwise. Find the derivative with respect to of (c) f(). Find the real and imaginary parts of the following functions: (a) f() + f() (c) f(). Determine whether the following functions are continuous at the origin:, (a) f() Re otherwise (a) ( ) (c) + ( ) 5. Evaluate the derivative of the function at the given point: (a) ( i), i, i 6. Find the derivative of the following functions: (a) f() a + b f() + (c) f() ( + ) (d) f() (e) f() + (f) f() + 7. Are the following functions analytic? (a) f() Im f() + (c) f() Re( ) (d) f() + (e) f() (f) f() 8. Find the most general analytic function f(x + iy) u(x, y) + iv(x, y) for which (a) u xy v xy (c) u e x cos y 9. Show that the following functions are harmonic, and find a corresponding analytic function f(x + iy) u(x, y) + iv(x, y) : (a) u x v xy (c) u xy (d) u x xy (e) v e x sin y
4 University College Cork: MA8 Complex Numbers and Functions 5 Exercises. Find a representation (t) of the line segment with endpoints (a), i + i, + 5i (c), 5 + i (d) + i, + 5i (e) i, 9 5i (f) i, 7 + 8i. Identify the curves represented by the following functions: (a) it : t ( + i)t : t (c) + i + e it : t < π (d) i e it (e) cos t + i sin t : π < t < π (f) t + it : t. Find a function (t) representing the following loci: (a) i + i (c) y x between (, ) and (, ) (d) y x between (, ) and (, ) (e) x + y (f) y + between (, ) x and (, 5 ). Integrate the function f() along the line segment running: (a) from to + i from to i (c) from + i to + 5i (d) from to i 5. Integrate the following functions: (a) along the parabola y x from (, ) to (, ) around the unit circle in the clockwise direction (c) from vertically to i and then horiontally to + i (d) along the straight line segment from to + i (e) a + b along the straight line segment from to + i. 6. Evaluate: d (a) clockwise around the C circle d anti-clockwise around the C circle (c) Im d anti-clockwise around C the circle r (d) d along the following curves C running from i to i : i. the straight line segment ii. the unit circle in the left half-plane iii. the unit circle in the right half-plane
5 University College Cork: MA8 Complex Numbers and Functions 5 Exercises 5. Integrate the following functions around the anti-clockwise unit circle, and in each case indicate whether Cauchy s Theorem may be applied: (a) f() f() e (c) f() (d) f() 5 (e) f() Im (f) f() Re (g) f() (h) f() (i) f() (j) f() + (k) f() tanh (l) f() sec. Given Im d π about the C anti-clockwise unit circle, use Cauchy s Theorem applied to the function f() to deduce the value of Re d about the same circle. C. Integrate the function f() along the following curves from to + i : (a) C runs vertically to i then horiontally to + i C is the straight line segment from to + i (c) C runs horiontally to then vertically to + i. (a) Show that +. Use the principle of deformation of path to show that d πi C where C is an anti-clockwise circle enclosing both the points and. 5. Integrate f() about the anti-clockwise circles (a) Could you obtain the second result from the first by the principle of deformation of path? 6. Evaluate the following integrals along the given curves: + (a) C d where C is the + anti-clockwise circle i. ii. iii. + C d where C is the clockwise boundary of the rectangle with vertices at ± i and ± i. 5
6 University College Cork: MA8 Complex Numbers and Functions 5 Exercises 6: Definite Integration; Cauchy s Integral Formula; Cauchy s Integral Formula for derivatives.. Evaluate the following integrals: (a) (c) (d) (e) +i i i i i i πi d ( + ) d ( + ) d ( ) d e d (f) (g) (h) (i) (j) πi π πi πi i i i i e d e / d e d sinh π d sin d (k) (l) (m) (n) (o) πi π πi πi i +i πi sin d cos d sinh π d e d cos d. Integrate the function f() + about the following anti-clockwise circles: (a) + i (c) i (d) i. Integrate the function f() about the following anti-clockwise circles: (a) + + (c) i (d). Integrate the following functions anti-clockwise about the unit circle: (a) (c) (d) cos + e e (e) i (f) ( π) cos (g) sin (h) e i 5. Integrate the following functions about the anti-clockwise circle : (a) (c) ( i) e π ( i) (d) (e) (f) (g) cos ( + ) sin π cos (h) (i) (j) e ( ) e e sin 6
7 University College Cork: MA8 Complex Numbers and Functions 5 Exercises 7: The Exponential Function; Trigonometric and Hyperbolic Functions; Logarithms.. Find (a) e πi/ e πi/ +i (c) e (d) e +5i. Express the following complex numbers in the form re iθ for some r and θ : (a) + i i. Find the real and imaginary parts of (a) e e (c) e (d) e e (c) The square roots of i and of i (d) The square roots of re iθ (e) The n th roots of re iθ. Find (a) cos sin (c) tan 5. Calculate (a) cos i cosh i (c) sin i (d) sinh( + i) 6. Show that (a) cosh cosh x cos y + i sinh x sin y (cosh ) sinh (c) cosh sinh (d) cos cosh i (e) sin i sinh i 7. Calculate the principal value of Ln for (a) + i i (c) 5 (d) i 7
8 University College Cork: MA8 Complex Numbers and Functions 5 Exercises 8: Taylor Series; Laurent Series.. Find the Taylor Series expansion about the given point a of each of the following functions and in each case determine the radius of convergence: (a) e, a, a (c) e, a i (d) cos, a π/ (e) sin, a π/ (f) sin π, a (g) cos( ), a (h) cos (), a (i) (j), a, a i. Expand the following functions as Laurent Series about the origin, and determine the precise region < < R of convergence: (a) sin ( ) (c) (d) e / 6 ( + ) (e) (f) 5 +. Find all Taylor Series and Laurent Series expansions of the following functions about the given point a, and determine the precise regions of convergence: (a) (c) +, a i +, a i, a (d) (e) (f), a ( ), a e ( ), a (g) (h) sin ( π ), a π, a i 8
9 University College Cork: MA8 Complex Numbers and Functions 5 Exercises 9: Zeroes and Singularities; Residues; The Residue Theorem.. Find the location and order of the eroes of the following functions: (a) ( ) (c) (9 + ) ( ) (d) (e) + i + ( + ) ( + ). Suppose that f() has a ero of order n at ζ, with n >. (a) Prove that f () has a ero of order n at ζ. Prove that has a pole of f() order n at ζ.. Find the location and type of each singularity of the following functions: (a) + e (c) 7 ( + ) (d) e /( ) (e) e /. Find the residues at the singular points of the following functions: (a) (c) e (d) e (e) 5 cos (f) ( ) (g) ( + πi) 6 e 5. Find the residue at each singular point which lies inside the circle : (a) (e) ( + )( + 6) (c) (d) 5 + ( ) 6. Evaluate the following integrals about the anti-clockwise unit circle: d d (a) C + (c) sin C (e) + 6i C d (g) d ( + ) + 9 (d) + d e (f) sin d (h) C C C C C e cos π d ( + ) d 7. Integrate + about the following anti-clockwise circles: (a) i i (c) i 8. Integrate (a) + ( )( ) about the following clockwise circles: (c) (d) 9
10 University College Cork: MA8 Complex Numbers and Functions 5 Exercises : Contour Integrals. Evaluate the following (real) integrals using methods of contour integration: (a) (c) (d) (e) dx x + dx ( + x ) + x + x dx dx ( + x ) x + x 8 dx (f) (g) (h) (i) (j) x (x x + ) dx dx x + x + 9 sin x x + x + dx cos x x + dx cos x (x + )(x + ) dx (k) dx (x + )(x + )(x + 9). Use elementary methods to find. Find a value for dx (x + )(x + ) dx x +. by integrating around the following contour: Im i R R Re. Integrate e around the boundary of the rectangle with vertices ±a, ±a + ib to show that π e b e x cos bx dx. Hint: Let a, and use e x dx π.
11 Exercises b i d 7 + i c xy x + y 5a (x + ) 5c + y (x ) + y 5e (x + y ) 5f (x + y ) 5h 5 6a (cos π/ + i sin π/) 6c 8(cos π + i sin π) 7a π 7c π/ c The region between the two branches of the hyperbola x y. e Horiontal strip of width π. g The circle (x 7 5 ) + y 6 5 i The left half-plane without the closed disk of radius and centre (, ). Exercises a ± + i c i, ± + i e ±5i + i g ± Exercises i ± + i, ± + i + i i k ±, ±i, ± m, ± i o ±, ±i a ±( + i) c ±( + i) a + i, i a + i, + i, i c 9 i i, i, 5 b (x xy ) x, (x y y ) y { b f() x y x + y, y, x. Hence not continuous. a 6( ) c 5b ( ) 7 7b Yes ( ) 7d No 7f Yes ( ) 8a i + ic, c real 8c e (cos y + i sin y) + ic, c real 9b 9d
12 Exercises a ( i)t : t c ( + i)t : t 5 e i + ( i)t : t a line segment from + i to i c Circle with centre + i, radius e Ellipse x + y Exercises 5 a i + e it : t π c t + it : t e cos t i sin t : t π a + i c 7 + i 5a 88 6i 5c 5 + 6i 6a πi 6b πi 6(d)i i 6(d)ii i 6(d)iii i a, no d, yes g, no k, yes b, yes e π, no h πi, no l, yes c, no f iπ, no i, no 6b πi Exercises 6 a + i c i e l i sinh π n e e a (by Cauchy s Theorem) c π/ a πi c πi e π/8 5c 5e 6πi 5g πi g c π g 5i j a πi/ 5a π 5j πi Exercises 7 a i c e(cos + i sin ) a e x cos y, e x sin y c e x xy cos(x y y ), e x xy sin(x y y ) a 5e i tan b e πi/ b sin x + sinh y 5a 5d approx i 7a πi ln i 7c ln 5 + πi.69 +.i
13 Exercises 8 a +!! +, R c e i ( + ( i) + ( i) + ),! R e! ( π ) +! ( π ) ), R f π π + π5 5, R! 5! + i j ( + + i ( ) + i ( i) + ( i) + ), R a c ( ) n n+ n, > (n + )! n n ( ) n, > n!n+6 e ( ) n n, < < n ( ) i n+ a ( i) n : n ( i) n < i < ; ( i) n+6 : i > c ( ) ( f n n : < ; n ) n n : > e( ) n : > n! n Exercises 9 a ±, ±i, simple d i,, simple e ±i, second order a (simple pole); ( nd order pole) c ±i ( rd order pole); (simple pole) a at c at e at f at ± 5a, i,, i at, i,, i 5c / at 5e 6a 7 6c π/ 6e at πi sin 6g i sinh 7a 6πi 7c 5πi 8b πi 8d Exercises b π/ d π/6 f π/ i πe / (sin + cos ) g π/ k π/6
14 Worked solutions to some of the exercise sheets Sheet 7 a b c d e pii/ cos π/ + i sin π/ i e πi/ cos π/ + i sin π/ i e +i e e i e(cos + i sin ) e cos + ie sin e +5i e e 5i e (cos 5 + i sin 5) e cos 5 + ie sin 5 a e e x iy e x e iy e x (cos y + i sin y) e x cos( y) + ie x sin( y) e x cos y i x sin y b e e (x+iy) e x y +ixy e x y e ixy e x y (cos xy + i sin xy) e x y cos xy + e x y sin xy c e e x +ix y xy iy e x xy ( cos (x y y ) + i sin (x y y ) ) d e e e ex+iy e ex (cos y+i sin y) e ex cos y+ie x sin y e ex cos y ( cos (e x sin y) + i sin (e x sin y) ) e ex cos y cos(e x sin y) + ie ex cos y sin(e x sin y) a + i + 5 arg + i tan (/) cos tan (/) + i sin tan (/) 5e i tan (/)
15 b i cos ( π/) + i sin ( π/) e iπ/ e iπ/ ; the square roots of i are e iπ/ and e πi/ d r e iθ/, r e i( θ +π) c The square roots of i are e iπ/ Sheet 9 a Simple eroes at ±, ±i and e n re i(θ+πk)/n for k,,..., n. a b Zero of order at c Double eroes when + 9, that is, when ±i. (Also triple poles at ±.) d Simple ero when i. (Also simple poles at ±i.) e Triple eroes when ±i. (Also a pole of order at.) a f() has a ero of order n at ζ if and only if f(ζ) f (ζ) f (n ) (ζ) f ( n)(ζ). Put g f and you see that g has a ero of order n. b f() has a ero of order n at ζ if and only if f(ζ) f (ζ) f (n ) (ζ) f (n) (ζ). This is the same as the condition for f to have a pole of order n at ζ. a Simple pole at b Double pole at c Triple poles at ±i d Isolated essential singularity at e Isolated essential singularity at f() Simple pole at ; res{f(); }. b Overt simple pole at ; res{f(); } (). c f() e g() with g() e. Pole of order at ; res{f(); } g ()!. d f() e h() k() has covert simple poles when e, that is, when πni, for n Z. res{f(); πni} h(πni) k (πni) 5
16 e Overt pole of order 5 at. [ ] d (cos ) d res{f(); }! f f() [cos ]! ( ) ( ) ( + )( ) ( + ) ( ) Double poles at ±. [ ] d res{f(); } d ( + ) res{f(); } e [ ( + ) ] [ ] d d ( ) ) g f() has an overt pole of ( + iπ) 6 order 6 at iπ. [ ] d 6 res{f(); iπ} d (e ) 5a f() eiπ 7 7 h() k() iπ with h(), k(), k (). There are covert simple poles at ±, ±i. 5b res{f(); } h() k () res{f(); } h( ) k ( ) res{f(); i} i i i i res{f(); i} ( i) ( i) i f() + + ( + )( + ) There are simple poles at,. res{f(); } ( ) ( ) res{f(); } ( ) ( )
17 5c simple poles at the cube roots of. f() 5 + ( + ) (overt) double pole at ; (covert) res{f(); } [ ] d d + [ ( + )( ) ( )( ] ) ( + ) 6 9 5d f() ( ) ( ) ( + ) ( i) ( + i) Double poles at ±, ±i. [ ] d res{f(); } d ( + ) ( + ) [ ( ( + )( + ) + ( + )()( + ) ) ] ( + ) ( + ) ( + ) (6 + ) e + f() ( + )( + 6) g() + where g() +. Thus f() has an + 6 overt simple pole at, with res{f(); } g( ) 7 (There are also (covert) simple poles at ±i.) 6a Set f() + and I f() d. (;) Then f() has (covert) simple poles when + ± i 7
18 and both of these points lie inside the circle. We have res{f(); i } 8(i/) i There are (overt) simple poles at, 6i ; the only pole inside the given circle is. res{f(); } 6i i 6 res{f(); i/} i i 8( i/) So (;) f() d πi i 6 π Hence I πi( i + i ). 6b f() poles at ± i. has (covert) simple + 9 6d ( + ) f() + ( + ) ( + ) Hence 6c res{f(); i/} i/ 8(i/) 8 res{f(); i/} i/ 8( i/) 8 (;) ( f() d πi 8 + ) 8 πi 8 πi 9 f() + 6i ( + 6i) Simple poles at (overt) and ± i (covert). res{f(); } 5 ( () + ) () + ( ) (i) + res{f(); i/} (i) 8(i) (i + ) 8 + i 8 i 8 ( ) ( i) + res{f(); i/} ( i) 8( i) i 8 8
19 (;) ( πi 5 + i + i ) 8 8 ( πi i ) 8 ( ) 9 πi i 9πi + π 6e f() sin at ±. (;) has covert simple poles res{f(); } sin 8( ) 6 sin res{f(); } sin 8( ) 6 sin ( f() d πi 6 sin + 6 sin ) πi sin 6f f() e has covert simple poles at sin πk ( k Z ). The only pole in the unit circle is. res{f(); } e cos (;) f() d πi πi 6g f() e has covert simple poles cos π at k + π ( k Z ). 7 res{f(); } e π sin π/ res{f(); /} (;) e/ π e / π sin( π/) e/ π ( ) e / f() d πi + e/ π π ( ) e / e / i i sinh f() + + ( ) ( + )( ) Simple poles at, ±. res{f(); } ( ) () + () res{f(); } ()() 8 9
20 () + ( ) res{f(); } ( )( ) 8 res{f(); } ()( ) 7a 7b (i;) ( f() d πi + + ) πi() 6πi 8a res{f(); } ()() (; ) f() d πi( ) πi 7c (+i;) (+i;) f() d πi πi ( ) f() d πi + 5πi + 8 f() ( )( ) poles at,,. res{f(); } has overt simple ( )( ) 8b 8c (; ) f() d πi πi (; ) f() d πi πi ( ) ( ) 8d f() d by Cauchy s ( ; ) Theorem. Sheet For these questions, I shall use the notation R to denote the (positively oriented) semicircular contour Γ R [ R, R]. (This is the contour I used in the two examples from the last lecture.) I shall also use I to denote the integral in the question and I R to denote the integral f() d of f() along the semicircular Γ R arc. I trust this won t be too confusing... a We integrate f() about the + semicircular contour R. f() has (covert) simple poles at ±i. res{f(); i} (i) i
21 So R f() d πi π ( ) i π f() d Rie it Γ R R e it + dt R R π R x + dx R as R R x + dx f() d f() d R Γ R π f() d Γ R π as R Hence I π. b f() ( + ) ( + i) ( i) There are (now overt) double poles at ±i. [ ] d res{f(); i} d ( + i) i [ ( + i) ] i (i) 8i i So R f() d πi π ( ) i π f() d Rie it Γ R ( + R e it ) dt O(R ) R as R f() d ( ) f() d f() d R Γ R π as R c Put f() +. There are covert + simple poles at cis ( π + kπ ) for k,,,, i.e., ± + i, ± i. res{f(); cis (π/)} [ ] + + cis (π/) cis (π/) cis (π/) ( + cis (π/))(cis ( π/) (cis ( π/) + cis ( π/)) ( i + i ) ( i) i
22 res{f(); cis (π/)} + cis (π/) cis (9π/) ( + cis (π/))(cis ( π/)) (cis ( π/) + cis (5π/)) i ( f() d πi i ) R π π π π f() d + R e it Γ R + R Rieit eit dt ( + R )(R) π R dt Hence R d + x + x dx R as R f(x) dx R ( ) f() d f() d R Γ R π f() d Γ R π f() as R ( + ) ( + i) ( i) Overt triple poles at ±i. res{f(); i} [ ] d! d ( + i) [ ] ( + i) 5 i [ ] (i) 5 [ ] i 6i i 6 So i R f() d Rie it Γ R ( + R e it ) dt O(R 5 ) R e f(x) hence f(x) dx R as R f(x) dx R ( ) f() d f() d R Γ R ( ( ) ) i πi I R 6 π 6 I R π 6 as R f(x) dx x is an odd function, + x8 f(x) dx f(x) dx + f(x) dx and f(x) dx I m not quite sure what the intention of this question was...
23 You can also show that this integral is ero by the much more convoluted route of finding the residue at each of the (simple) poles inside the standard semicircular contour, and discovering that they all add up to ero. f + ( ) +, so f() ( ) has covert double poles + when ( ), that is, ± i. To find the residues, we need to express f() in such a way as to make the poles overt: f() ( + ) ( ( ) + ) ( ( ) + i ) ( ( ) i ) ( ( i) ) ( ( + i) ) using a standard Estimation Theorem argument on I R. g f() ( + )( + 9) Covert simple poles at ±i, ±i. res{f(); i} (i) + 9 (i) i 8 i 6 res{f(); + i} [ ] d ( ) d ( i) +i [( ) ( )] ( i) ( i) ( ) ( i) (i) ( + i)(i) (i) i + 6 i R x R x x + ) dx f() d f() d R Γ R πi i I R π as R +i R res{f(); i} (i) + (i) 6i 8 i 8 R f(x) dx f(x) dx R f() d f() d R Γ R ( πi i ) I R πi ( i 8 π I R 6 + i 8 ) I R π as R
24 Solutions to Sheet a ± + i g + i (x + iy) + i b ± i c i cis (π/). cis (π/6), cis (5π/6), cis ( π/) + i, + i, i. d i cis (π/). cis (π/8), cis (5π/8), cis ( π/8) or cis ( 7π/8). x y xy y x x 6x 6x 8x 6x 8x (x )(x + ) x (because x ) x ± y (± /) ± + i ± h + i cis (π/). cis (π/), cis (π/), cis (7π/). e ±5i i cis π. cis (±π/), cis (±π/) ± + i, ± i. j cis. cis (πk/7) for k,..., 6 or for k,...,. f 8 cis. cis (kπ/) for k,,...,. ±, ±i, ± + i, ± i k 6 cis π. cis ( π 6 + πk ) for k,..., 5 or for k,...,. cis (±π/6), cis (±π/), cis (±5π/6) + i ±, ±i, ± + i
25 Solutions to Sheet Solutions to Sheet a (t) ( i)t : t b (t) t( + 5i) + ( t)( + i) t + 5it + i + t it t + i(t + ) (t ) c (t) ( + i)t : t 5, or (t) (5 + i)t : t d e (t) ( + 5i)t + ( t)( + i) t + 5it + + i t it t + + i(t + ) ( t ) (t) (9 5i)t + ( t)( i) 9t 5it + i t + it 8t + i(t + ) ( t ) or (s) s + i(s + ) i + ( i)s for s. f (t) ( 7 + 8i)t + ( t)( i) 7t + 8it i + it 7t + i(t ) ( t ) a Straight line segment from + i to i b Straight line segment from to + 6i c Anticlockwise circle, centre + i, radius, traced from the positive real direction. d Anticlockwise circle, centre i, radius, traced from the negative real direction. e Put x cos t, y sin t. Then cos t + sin t ( y x + ) a i + e it ( t π) b i + e it ( t π) c t + it ( t ) d t + t e ( t ) x + y ( x ) + y Put x cos t, y sin t. Then cos t + i sin t ( t π). 6c f() Im (t) e it (for t π ). f() d π π cos t + i sin t (t) ie it i cos t sin t sin t(i cos t sin t) dt (i sin t cos t sin t) dt π [ ( )] cos t i sin t cos t dt [ i sin t t ] π sin t + π x + y This is the equation of the (anticlockwise) ellipse traced out. 5
26 Solutions to Sheet 5 b f is holomorphic everywhere, so Cauchy s Theorem applies and the integral is ero. d f is holomorphic inside and on the given circle (the only singularity is outside the curve), so Cauchy s Theorem does apply, and the integral is ero. f (;) Re d π π iπ cos t(i cos t sin t) dt (i cos t sin t cos t) dt h i (;) π i i d π π (cos t i sin t)(i cos t sin t) dt ( cos( t) + i sin( t) )( cos t + i sin t ) dt dt πi (;) d (;) (;) (;) Re d i( π) iπ π ( i Im ) d (i cos t sin t) dt d i Im d (;) +i [ d ] +i ( + i) + i i i (i ) (Note: This only works because the integrand is antidifferentiable; if you calculate the different path integrals you should get the same answer.) 6
27 5a (;) π d i πi π e it (ieit ) dt dt 5b (;) π d 8π π e it ie it dt dt 5 The principle of deformation of path does not apply because the integrand is not holomorphic. Solutions to Sheet 6 Note: The integrands in these questions are all holomorphic (and also integrable) everywhere, and consequently the given integrals are independent of the path used to get from the initial point to the final point. We thus use the Fundamental Theorem of Calculus for complex functions. b a +i [ d ] +i ( + i) + i i + i i i [ ( + ) (x + ) d ] i i (i + ) (i + ) 7i.9. +.i i.. +.i ( 7i 8 + i i ) ( ) i(5 96 7) (6 + 7i) + 9i 7
28 c i [ ( + ) d + i + i i d Warning: There is a mistake somewhere in this answer. It should come out to i Kudos to the first person who can provide a correct solution. i i ( ) d i i ( 6 + ) d [ ] 7 i i [ 8i.i ] [ i 8i i i ] 5 i i 8i 96i 7 5 i + i 7 + i 5 i ( i ) 5 i 76 5 ] i e g πi e d [e ] πi e πi e πi πi [ e /] πi πi [ e πi] [ e πi] ( )( ) ( )( ) f πi π [ e e d ] πi π e6πi e π e π h i e d [ ] i e e e (e e ) sinh 8
29 i j i i i [ cosh π sinh π d π cosh πi π ] i eπi + e πi π π π π sin d [ cos ] i i cosh π π cos i + cos( i) cosh + cosh( ) (because cosh is an even function) k l πi π [ ] cos πi sin d π cos πi cos π + cosh π eπ + e π eπ + + e π + ( e π + e π ) (cosh π) πi πi cos d [sin ] πi πi sin πi sin( πi) sin πi i sinh π i(e π e π ) m n o i [ cosh π sinh π d π cosh πi π +i πi e d cos π π π [ π ] e +i ] i cosh π e e+i 6 i+ e e e e ] πi [ sin cos d sin( π ) sin π f() + ( + i)( i). Simple poles at ±i. Unless otherwise stated, these answers involve Cauchy s Integral Formula. a (; ). f() is holomorphic inside and on, so f() d by Cauchy s Theorem. b ( i; ). [ ] f() d πi i ( ) πi i π i 9
30 c (i; ). [ ] f() d πi + i i πi π d (i; ) [ ] f() d πi + i i π ( ) i (You could also use the Principle of Deformation of Path to obtain this result from the previous one.) f() ( + )( )( + i)( i) Again we use Cauchy s Integral Formula unless otherwise indicated. a ( ; ) f() d [ ] πi ( )( + i)( i) πi ( )() πi b ( ; ). f() d by Cauchy s Theorem. c (i; ) [ ] f() d πi ( + )( )( + i) πi ( )(i) π i d (; ) [ ] f() d πi ( + )( + i)( i) πi ()() πi Again we use Cauchy s Integral Formula, with (; ). a b cos (Cauchy s Theorem) c d e d πi cos πi + d e e d πie πi d πi(e ) i d i d πi [ ] i/ ( ) i πi 8 π 8
31 f (Cauchy s Theorem) g h cos π d sin d sin d πi sin e i d e i d πi e(i/) πiei 5 (; ) ; these questions use Cauchy s Integral Formula for Derivatives. 5a [ ] πi d d ( i)! d i πi.i i 5d 5e 5f 5g 5h cos [ ] d d πi d cos πi( sin ) [ ] πi d d ( + )! d πi.6( ) sin π cos πi! 6πi [ ] d sin π d πi( π ) sin d πi! πi [ ] d d cos [ ] e d d πi ( ) d e πie 5b 5c e π [ ] πi d d! d eπ πi.πe (Cauchy s Theorem) π i ( i) d 5i 5j e sin [ ] e πi d d! d e [ ] d d πi d e sin πi [e cos + e sin ] πi
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 περισσότεραF19MC2 Solutions 9 Complex Analysis
F9MC Solutions 9 Complex Analysis. (i) Let f(z) = eaz +z. Then f is ifferentiable except at z = ±i an so by Cauchy s Resiue Theorem e az z = πi[res(f,i)+res(f, i)]. +z C(,) Since + has zeros of orer at
Διαβάστε περισσότεραSection 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 περισσότεραCHAPTER 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 περισσότεραMath221: 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 περισσότεραFourier 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 περισσότεραSection 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 περισσότεραCOMPLEX NUMBERS. 1. A number of the form.
COMPLEX NUMBERS SYNOPSIS 1. A number of the form. z = x + iy is said to be complex number x,yєr and i= -1 imaginary number. 2. i 4n =1, n is an integer. 3. In z= x +iy, x is called real part and y is called
Διαβάστε περισσότεραAreas 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 περισσότεραAreas 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 περισσότεραExample 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 περισσότερα2 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 περισσότεραHomework 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 περισσότεραInverse 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 περισσότεραb. Use the parametrization from (a) to compute the area of S a as S a ds. Be sure to substitute for ds!
MTH U341 urface Integrals, tokes theorem, the divergence theorem To be turned in Wed., Dec. 1. 1. Let be the sphere of radius a, x 2 + y 2 + z 2 a 2. a. Use spherical coordinates (with ρ a) to parametrize.
Διαβάστε περισσότεραPractice 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 περισσότεραSolutions 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 περισσότεραMatrices 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 περισσότεραCRASH 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 περισσότεραTrigonometric 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 περισσότεραSection 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 περισσότεραExercises 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 περισσότεραANSWERSHEET (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 περισσότεραHomework 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 περισσότεραSecond 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 περισσότεραConcrete 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 περισσότεραD 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 περισσότεραChapter 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 περισσότεραEcon 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( 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 περισσότεραCHAPTER 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 περισσότεραFinite 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 περισσότεραTRIGONOMETRIC FUNCTIONS
Chapter TRIGONOMETRIC FUNCTIONS. Overview.. The word trigonometry is derived from the Greek words trigon and metron which means measuring the sides of a triangle. An angle is the amount of rotation of
Διαβάστε περισσότεραPg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
Διαβάστε περισσότεραQuadratic 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 περισσότεραHOMEWORK 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 περισσότεραEE512: 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 περισσότεραProblem 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 περισσότεραPARTIAL 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 περισσότεραIf 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 περισσότεραNumerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Διαβάστε περισσότεραChapter 1 Complex numbers
Complex numbers MC Qld- Chapter Complex numbers Exercise A Operations on and representations of complex numbers a u ( i) 8i b u + v ( i) + ( + i) + i c u + v ( i) + ( + i) i + + i + 8i d u v ( i) ( + i)
Διαβάστε περισσότεραIf 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 περισσότεραParametrized 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 περισσότεραΚΥΠΡΙΑΚΗ 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 περισσότεραDifferential equations
Differential equations Differential equations: An equation inoling one dependent ariable and its deriaties w. r. t one or more independent ariables is called a differential equation. Order of differential
Διαβάστε περισσότερα10/3/ revolution = 360 = 2 π radians = = x. 2π = x = 360 = : Measures of Angles and Rotations
//.: Measures of Angles and Rotations I. Vocabulary A A. Angle the union of two rays with a common endpoint B. BA and BC C. B is the vertex. B C D. You can think of BA as the rotation of (clockwise) with
Διαβάστε περισσότεραSecond 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 περισσότεραderivation 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 περισσότεραSimilarly, we may define hyperbolic functions cosh α and sinh α from the unit hyperbola
Universit of Hperbolic Functions The trigonometric functions cos α an cos α are efine using the unit circle + b measuring the istance α in the counter-clockwise irection along the circumference of the
Διαβάστε περισσότεραReminders: 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 περισσότεραMath 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 περισσότεραSrednicki 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 περισσότεραSCHOOL 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 περισσότεραSpherical 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 περισσότεραApproximation 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 περισσότεραAnswer sheet: Third Midterm for Math 2339
Answer sheet: Third Midterm for Math 339 November 3, Problem. Calculate the iterated integrals (Simplify as much as possible) (a) e sin(x) dydx y e sin(x) dydx y sin(x) ln y ( cos(x)) ye y dx sin(x)(lne
Διαβάστε περισσότεραC.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 περισσότεραThe 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 περισσότεραReview Test 3. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Review Test MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. 1) sin - 11π 1 1) + - + - - ) sin 11π 1 ) ( -
Διαβάστε περισσότεραk 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 περισσότεραPhys460.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 περισσότεραforms 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 περισσότεραChapter 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 περισσότερα4.6 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 περισσότερα9.09. # 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 περισσότεραTrigonometry 1.TRIGONOMETRIC RATIOS
Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y
Διαβάστε περισσότεραSCITECH 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 περισσότεραLecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
Διαβάστε περισσότεραDESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Διαβάστε περισσότεραNowhere-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 περισσότεραLecture 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 περισσότεραΑπόκριση 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 περισσότεραECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Διαβάστε περισσότερα1. (a) (5 points) Find the unit tangent and unit normal vectors T and N to the curve. r(t) = 3cost, 4t, 3sint
1. a) 5 points) Find the unit tangent and unit normal vectors T and N to the curve at the point P, π, rt) cost, t, sint ). b) 5 points) Find curvature of the curve at the point P. Solution: a) r t) sint,,
Διαβάστε περισσότεραStrain 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 περισσότεραSolution 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 περισσότεραRectangular Polar Parametric
Harold s Precalculus Rectangular Polar Parametric Cheat Sheet 15 October 2017 Point Line Rectangular Polar Parametric f(x) = y (x, y) (a, b) Slope-Intercept Form: y = mx + b Point-Slope Form: y y 0 = m
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραDifferentiation exercise show differential equation
Differentiation exercise show differential equation 1. If y x sin 2x, prove that x d2 y 2 2 + 2y x + 4xy 0 y x sin 2x sin 2x + 2x cos 2x 2 2cos 2x + (2 cos 2x 4x sin 2x) x d2 y 2 2 + 2y x + 4xy (2x cos
Διαβάστε περισσότεραJackson 2.25 Homework Problem Solution Dr. Christopher S. Baird University of Massachusetts Lowell
Jackson 2.25 Hoework Proble Solution Dr. Christopher S. Baird University of Massachusetts Lowell PROBLEM: Two conducting planes at zero potential eet along the z axis, aking an angle β between the, as
Διαβάστε περισσότεραDerivations of Useful Trigonometric Identities
Derivations of Useful Trigonometric Identities Pythagorean Identity This is a basic and very useful relationship which comes directly from the definition of the trigonometric ratios of sine and cosine
Διαβάστε περισσότεραPaper Reference. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced. Thursday 11 June 2009 Morning Time: 1 hour 30 minutes
Centre No. Candidate No. Paper Reference(s) 6665/01 Edexcel GCE Core Mathematics C3 Advanced Thursday 11 June 2009 Morning Time: 1 hour 30 minutes Materials required for examination Mathematical Formulae
Διαβάστε περισσότεραOrdinal 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 περισσότεραDiracDelta. 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 περισσότεραAppendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Διαβάστε περισσότεραJesse 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 περισσότεραIntegrals 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 περισσότεραST5224: 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 περισσότεραMock 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 περισσότεραCHAPTER 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 περισσότεραStatistical 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 περισσότεραUniform 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 περισσότεραPartial 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 περισσότεραArithmetical 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 περισσότερα2 2 2 The correct formula for the cosine of the sum of two angles is given by the following theorem.
5 TRIGONOMETRIC FORMULAS FOR SUMS AND DIFFERENCES The fundamental trignmetric identities cnsidered earlier express relatinships amng trignmetric functins f a single variable In this sectin we develp trignmetric
Διαβάστε περισσότεραSection 8.2 Graphs of Polar Equations
Section 8. Graphs of Polar Equations Graphing Polar Equations The graph of a polar equation r = f(θ), or more generally F(r,θ) = 0, consists of all points P that have at least one polar representation
Διαβάστε περισσότεραProblem 1.1 For y = a + bx, y = 4 when x = 0, hence a = 4. When x increases by 4, y increases by 4b, hence b = 5 and y = 4 + 5x.
Appendix B: Solutions to Problems Problem 1.1 For y a + bx, y 4 when x, hence a 4. When x increases by 4, y increases by 4b, hence b 5 and y 4 + 5x. Problem 1. The plus sign indicates that y increases
Διαβάστε περισσότερα1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Διαβάστε περισσότεραNew 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