Department of Mathematics and Applied Mathematics

Course: Complex Analysis: MAM 3000W - CA

Tutorial N5

August 23, 2013

1. Expand the function f(z) =1

z − ain power series around z = 0 and

z = 1 (we assume that a 6= 0; 1). What are the radii of convergence ofthese series?

2. Prove that | sin z|2 + | cos z|2 = 1 if and only if z is real.

3. Define tan z = (sin z)(cos z)−1. Where this function is defined and ana-lytic?

4. Show that the system of two real differential equations:

d

dtp = −ωq,

d

dtq = ωp, ω = const , ω 6= 0

is equivalent to one complex differential equation for z(t) = p(t) + iq(t).Find the general solution of that equation and the general solution ofthe above system.

5. Suppose that f : G 7→ C is a branch of the logarithm and n is an integer.Prove that for z ∈ G zn = exp(nf(z)).

6. Let f be an analytic function, defined on the disk B(0; 1).

• Prove that the function φ(z) = f(z̄) is also analytic.

• Prove that the function ψ(z) = f(z) can be analytic only in thecase f(z) = const .

7. Let F (z) = z3. Prove, that there does not exist a point c on the linesegment [1, i] (the segment joining 1 and i as points on the complexplane) such that

F (i)− F (1)

i− 1= F ′(c).

(The exercise shows that the Mean Value Theorem is not true in thecomplex case).

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