1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

584 ANSWERS 25c. Letting z1 = 2i, and z2 = - 2i, we compute K = 13 + 2i - 2i l + 13 + 2i + 2il = 3 + 5 = 8. Then, with z = (x, y ...

ANSWERS 585 Section 1.5. The Algebra of Complex Numbers, Revisited: page 37 la. - 16 - il6J3. le. -64. cos39 = cos^3 9 - 3 cos ...

586 ANSWERS 13. Let C: z(t) = (x (t), y (t)), as ts b be any curve joining -2 and 2. Then x (a)= - 2, and x (b) = 2. By the inte ...

The triangle with vertices - 5 - 2i, -6, and 3 + 2i. 15a. w = f (z) =^3 tJi z +^7 if. 15 c. w = f (z) = !z+^7 ~^4 i. ANSWERS 5 ...

588 ANSWERS T he right half-plane given by Re(z) > 0 is mapped onto the region in the right half-plane satisfying u^2 - v^2 ...

ANSWERS 589 Make use of standard techniques. For example, to show that f + g is con- tinuous, use T heorem 2.2 applied to the s ...

590 ANSWERS 15. T he intersection of D~ (~) = { (u, v): (u- !) 2 + v^2 < ~} and Dt (-~) = {Cu, v): u^2 +(v+ ~) 2 < !}· Th ...

ANSWERS 591 ll. We evaluate lim f(• ) - / (O) = lim l•l'-o = lim •• = lim z = 0. Follow the z - O z-0 z - 0 .z-0 z-0 z z-0 hint ...

S92 ANSWERS Section 3.3. Harmonic Functions: page 119 l a. u is harmonic for all values of (x, y). 3. c = - a. Sa. v(x,y)=x^3 - ...

ANSWERS 593 00 00 I:; (a+ ib) (xn + iyn) = I:: [(axn - by,.)+ i(bxn + ayn)]. By Theorem 4.4, n = l n=l this expression equals ...

594 ANSWERS (show t he details for this). But this last quantity equals J3 -v'6 (explain), which is less than 1 (again, explain) ...

3c. ~- 3e. l· 3g. ~- 3i. 1. ANSWERS 595 5.Showthat limsuplc;,I • = l. ( limsuplenl;; ')2. n-oo n - oo 00 7. The theorem establis ...

S9 6 ANSWERS 7c. z = ln2 + i (-~ + 2mr), where n is an integer. 9a. exp(z) = f ;h (zt = Jim ( f, ~ (zt) = lim n = O k-oo n=O k-o ...

ANSWERS 597 5 e. Note that z^2 - z + 2 = 0 when z = i±;Y!. Also, Arg ( l±;Y!) = ±Arc tan ( J7) , respectively. For a = Arc tan ( ...

598 ANSWERS (^3) · t an z -_ 2( cos (^2) x cosh^2 cosxsl (^2) y+slnn x (^2) x sinh (^2) y) + i • 2 (cos• x cosh2cos 2 h 11sinhy ...

ANSWERS 599 lk. i (! + n) n, where n is an integer. Section 6.1. Complex Integrals: page 197 la. 2 - 3i. le. 1. le. J21T' /8 + V ...

600 ANSWERS 19. The absolute value of t he integrand is .Jx^2 + (1 - x^2 ) cos^2 rl, which simpli- fies to .J x^2 sin^2 8 + cos^ ...

ANSWERS 601 lnv'2-t+i(lnv'2+~-1). Log (1 + i) - Log (2) + Log (2 + i) =-~In J2 +In J5 + i (~+Arc tan 4). 15. Parametrize C ...

602 ANSWERS We know that the complex cosine is an entire function that is not a constant. By Liouville's theorem, it is not bou ...

ANSWERS 603 there exists an integer N. such that n 2': N. implies lln (z) - I (z)I < ~ for all z E T. Likewise, there exists ...