z2 z

z3

z

z

|z|

Re z

e z

cosh z

sin z

1 z

2z + 3 z + 2

2z i iz + 2

z z2 + 1

ln |z|

z2 z

ez

z2 z 2

cos z

z

ln z (Use 0 << 2.)

(1 + 2i)z2 + (i 1)z + 3

eiz (Careful: cos z and sin z are not u and v.)

y + ix

x iy x2 + y2

y ix x2 + y2

d dz [Af(z) + Bg(z)] = A df dz + B dg dz .

d dz [f(z)g(z)] = f(z) dg dz + g(z) df dz .

d dz f(z) g(z) = gf fg g2 , g(z) = 0. 28

d dz f[g(z)] = df dg dg dz . (See hint below.)

d dz (z3 )=3z2 . 3

d dz (zn) = nzn1 .

d dz ln z = 1 z , z = 0. Hint: Expand ln 1 + z z in series. 3

Using the definition of ez by its power series [(8.1) of Chapter 2], and the theorem (Chapters 1 and 2) that power series may be differentiated term by term (within the disk of convergence), and the result of Problem 30, show that (d/dz)(ez) = ez.

Using the definitions of sin z and cos z [Chapter 2, equation (11.4)], find their derivatives. Then using Problem 27, find (d/dz)(cot z), z = n.

ln(1 z)

cos z

p1 + z2

tanh z

1 2i + z

z z2 + 9

(1 z) 1 4

eiz

sinh z

In Chapter 12, equations (5.1) and (5.2), we expanded the function (x, h) in a series of powers of h. Use Theorem III (see instructions for Problems 34 to 42 above) to show that the series for (x, h) converges for |h| < 1 and 1 x 1. Here h is the variable and x is a parameter; you should find the (complex) value of h which makes infinite, and show that the absolute value of this complex number is 1 (independent of x when x2 1). This proves that the series for real h converges for |h| < 1.

Prove Theorem IV, Part 1. Hint: Recall the equality of the second cross partial derivatives; see Chapter 4, end of Section1.

Let f(z) = u+iv be an analytic function, and let F be the vector F = vi+uj. Show that the equations div F = 0 and curl F = 0 are equivalent to the Cauchy-Riemann equations.

Find the Cauchy-Riemann equations in polar coordinates. Hint: Write z = rei and f(z) = u(r, ) + iv(r, ). Follow the method of equations (2.3) and (2.4).

Using your results in Problem 46 and the method of Problem 44,

z

|z|

ln z

zn

z| 2

z| 1/2 e i/2

y

3x2 y y3

xy

x + y

cosh y cos x

e x cos y

ln(x2 + y2 )

x x2 + y

ey sin x

y (1 x)2 + y2

It can be shown that, if u(x, y) is a harmonic function which is defined at z0 = x0 + iy0, then an analytic function of which u(x, y) is the real part is given by f(z)=2u z + z0 2 , z z0 2i + const. [See Struble, Quart. Appl. Math., 37 (1979), 79-81.] Use this formula to find f(z) in Problems 54 to 63. Hint: If u(0, 0) is defined, take z0 = 0.

p (p + 1)(p2 + 4)

In equation (7.18), let u(x) be an even function and v(x) be an odd function. (a) If f(x) = u(x)+iv(x), show that these conditions are equivalent to the equation f(x) = f(x). (b) Show that u(a) = P V Z 0 2xv(x) x2 a2 dx, v(a) = P V Z 0 2au(x) x2 a2 dx.