1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

604 ANSWERS h (•)(<>) 13! (•)(<>) 13. To verify Identity (7- 15 ), let h(z) = f3f (z). Clearly, ----;ir-= n! = 00 f3 ...

ANSWERS 605 00 15b. f (z) = l + l L 2-n (zn + z - n). n = l 00 17. Since f (z) = L anzn is valid for lzl = 1 (explain), letting ...

606 ANSWERS 3e. Removable singularity at the origin, and a simple pole at -1. 3g. Removable singularity at the origin. By Theor ...

Section 8.1. Calculation of Residues: page 299 la. 1. le. 1. le. 1. lg. o. li. e. lk. o. lm. 4. 3a. ,,. 1 i" 3c. (1 - cos 1) 2 7 ...

608 ANSWERS 5 1l' u. s · 13. 3.,, 8. Section 8.3. Improper Integrals of Rational Functions: page 310 o. 5.^7 TS1f · 7. ~· 9. ~ ...

~· 1r. ~· 7r (1-~). 13. 1f (1 - cos 1). 15. 'Ir. Section 8.6. Integrands with Branch Points: page 326 (^1). v'321'. nln2. (^ ...

610 ANSWERS 3. 3[xn] = L::°=O XnZ-n = 0 +Oz-I+ Oz-Z + ... + Ozm-l + L::"=m. lz- n, 3[ J m"oo ( -l)n - m 1 z - m z . •-m Xn = ...

ANSWERS 611 1 3a. Take the z-transform of both sides z(Y(z) - 1000) = (1 - ~)Y(z). Solve for Y(z) and get Y(z) = ~~!', then find ...

612 ANSWERS residues of f(z) = Y(z)zn- l at the poles Res[f(z), 3 +ii = limz ..... 3+i(z - 3 - ') i (z - J +2.2-Sz i)(z- 3-i) z ...

ANSWERS 613 Section 9.3. Digital Signal F ilters: page 390 la. Substitute x(n] = cos(23 n) and get y[n] = x(n] + x(n - l] + x[n ...

614 ANSWERS 9. Recall that the solutions to z^6 = 1 are the sixth roots of unity z = e-'¥ for k = 0,1,2,3,4,5 and lie on the uni ...

ANS\VERS 615 Sect ion 10. 1. Basic Properties of Conformal Mappings: page 401 la. All z. le. All z except z = - 1. l e. All z ex ...

616 ANSWERS 17. The equation z = ~=~~ can be written as cz^2 + (d - a)z - b = 0, and a quadratic equation has, at most, two dist ...

<fi(x,y) = 4 - ~Arg(z + 3) + *Arg(z + 1) - ~Arg(z-2). <f!(x,y) = 4 - ~Arctan~ + ~Arctanrl:T - ~Arctant-:2. (^7) ·"'(x '+ ...

6 1 8 ANSWERS 11. T(x, y) =^2 ~^0 Re(Arcsin±). 1 • 2 1 3. Isothermals are T (x, y) = k. The equation 100 -^1 ~^0 arctan -:i: 2 Y ...

ANSWERS 619 Section 11.9. The Schwarz-Christoffel Transformation: page 492 f'(z) = A(z-a)-(or-kor)/" = A(z-a)k-i, integrate and ...

620 ANSWERS W = 2(z + l)~ + 1-0gl-(z+ l)! + i7r. 1+(%+1)! (^9) • w -- l tr Arcsinz + i fr Arcsin .! ~ + !±i 2' Section 1 2.l. ...

ANSWERS 621 Section 12.3. Vibrations in Mechanical Systems: page 535 (^00) (-1rs (^00) (-1jA(n (^2) - 1). la. Up (t) = L:: n•+ 7 ...

622 ANSWERS Section 12 .6. Laplace Transforms of Derivatives and Integrals: page 552 1. .C (sint) = 8 2~ 1. 3. .C (sin^2 t) = s( ...

ANSWERS 623 Section 12.8. Multiplication and D ivis io n b y t: page 561 1. [, (te-2t) = (s.1 2 )2. 3. .C (tsin3t) = (• 2 6.;. 9 ...