MATH 396 PROBLEMS 10

IGOR KRIZ

Regular problems:

1. Calculate the integral of the function

f(z) = \frac{3z − 4}{z^{2} − 3z + 2}

over (a) the circle with center 0 and radius 3/2 oriented counter-clockwise

(b) the circle with center 0 and radius 3 oriented clockwise. [Use the method of partial fractions.]

2. Are the following functions holomorphic?

(a) f(x + iy) = cos(x^{2} + y^{2})(e^{2xy} + e^{−2xy}) + i sin(x^{2} + y^{2})(e^{−2xy} − e^{2xy}).

(b) g(x + iy) = e^{(e^{x} cos y)} cos(e^{x} sin y) + ie^{(e^{x} cos y)} sin(e^{x} sin y).

3. A holomorphic function f : ℂ − {0} → ℂ has the property that ∫_{L} f = 0 for any circle L in ℂ − {0} with radius 1. Is f necessarily constant? (Give a reason.)

4. Calculate (from first principles) the complex derivatives of all the functions z^{n}, n ∈ ℤ.

Challenge problem:

5. Let p(z) be a polynomial with complex coefficients which has no roots of higher multiplicities. Prove that if L is a piecewise smooth Jordan curve oriented counter-clockwise, then the number of roots of p which are located inside L is equal to

\frac{1}{2πi} ∫_{L} \frac{p′(z)}{p(z)}.

[First realize that the integral is 0 around areas in which p has no roots. Thus, it suffices to consider the case when L is a very small circle inside of which there is exactly one root x. In this case, write p(z) = (z − x)q(z) where q has no roots inside L. Rewrite the integral in terms of q, and use the facts we already know.]