1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

(e) f(z)=x^3 +i(l- y)^3.

(f) f(z) = z^2 +z.

(g) f (z) = x^2 + y^2 + i2xy.

(h) f (z) = lz - (2 + i)l^2.

3.2 • THE CAUCHY-RIEMANN EQUATIONS 111

2. Let f be a differentiable function. Verify t he identity I f ' (z)l^2 = u; +v; = ~ +v;.

3. Find t he constants a and b such that f (z) = (2x- y)+i(ax + lnJ) is differentiable

for all z.

4. Let f be differentiable at zo = r 0 e'^90 • Let z approach zo along t he ray r > 0,

(}=Bo and use Equation (3-1) to show that Equation (3- 14) holds.

5. Let f (z) = e"' cosy+ ie"' sin y. Show that both f (z) and f' (z) are differentiable
   for all z.


6. A vector field F (z) = U (x, y) + iV (x, y) is said to be irrotational if Uv (x, y) =
   V., (x,y). It is said to be solen oidal if U., (x,y) = - V 11 (x,y). If f (z) is an analytic
   function, show that F (z) = f (z) is both irrotational and solenoidal.


7. Use any method to show that the following functions are nowhere differentiable.

(a) h (z) = e^11 cosx + ie^11 sin x.

(b)g(z)=z+z.

8. Use Theorem 3.5 with regard to the following.

(a) Let f (z) = f (rei^9 ) = In r +iii, where r > 0 and -'Ir < Ii < 1 r. Show

that f is analytic in the domain indicated and that/' (z) = ~·

(b) Let f (z) = (In r)^2 - 02 + i211 In r, where r > 0 and -'Ir <Ii $'Ir. Show

that f is analytic for r > 0, -'Ir<(}< 'Ir, and find!' (z).

9. Show that the following functions are entire (see Definition 3.1).

(a) f (z) = coshxsiny- isinhxcosy.

(b) g (z) = cooh x coo y + isinh x sin y.

10. To prove T heorem 3.5, the polar form of t he Cauchy- Riemann equations,

(a) Let f (z) = f (x, y) f (rei^9 ) = u (re^18 ) + iv (re;^8 ) =

U (r, B) + iV (r, 0). Use the transformation x = r cos(} and y = r sin(}

(i.e., (x, y) = r e^18 ) and the chain rules

Ur = ux ~: +Uy: and Ue = u., ~: +Uy~~ (similarly for V)

to prove that

Ur = Ux cos B + u 11 sin(} and Ue = -u,r sin B + u 11 r cos Ii and

Vr = v., cos Ii + v 11 sin Ii and Ve = -v, r sin Ii + vvr cos Ii.