MAT 542 Midterm Examination Solutions

1. (20 pts)

(a) \underline{Define} the term ”radius of convergence” of the power series ∑_{n=0}^{∞} a_{n}z^{n}. SOL’N: The radius of convergence of ∑_{n=0}^{∞} a_{n}z^{n}

= sup{r ≥ 0; ∑_{n=0}^{∞} |a_{n}|r^{n} < +∞}.

(b) Determine the radius of convergence of ∑_{n=0}^{∞} 2^{n}z^{n^{2}}. SOL’N: Use the ratio test and calculate

lim_{n→∞} \frac{2^{n+1}|z|^{(n+1)^{2}}}{2^{n}|z|^{n^{2}}} = lim_{n→∞} 2|z|^{2n+1} = \{ \begin{matrix}0 & for |z| < 1 \\ 2 & for |z| = 1 \\ ∞ & for |z| > 1 \end{matrix} .

Hence the radius of convergence of this series is 1.

Give an example of a power series with radius of convergence 1 that

(c) converges for all |z| ≤ 1, SOL’N: ∑_{n=0}^{∞} \frac{1}{n^{2}} z^{n}.

(d) converges at some but not all points z with |z| = 1, SOL’N: ∑_{n=0}^{∞} \frac{1}{n}z^{n} converges at z = −1 and diverges at z = 1.

(e) diverges for all z with |z| = 1. SOL’N: ∑_{n=0}^{∞} z^{n}.

2. (30 pts) Let f = u + ıv be a C^{2}-function on ℂ (as usual, u = ℜf and v = ℑf).

(a) What are the Cauchy-Riemann equations for u and v (in terms of the real partial derivatives with respect to x and y)? SOL’N:

u_{x} = v_{y} and u_{y} = −v_{x}.

(b) What is the Cauchy-Riemann equation for f (in terms of the complex partial derivatives with respect to z and \overline{z})?

SOL’N:

f\overline{z} = 0.

(c) Determine the constants A, B and C such that

f_{x x} + f_{y y} = Af_{z z} + Bf_{z \overline{z}} + Cf_{\overline{z} \overline{z}}.

SOL’N:

f_{x} = f_{z}z_{x} + f\overline{z}\overline{z}_{x} = f_{z} + f\overline{z}

and

f_{y} = f_{z}z_{y} + f\overline{z}\overline{z}_{y} = ı (f_{z} − f\overline{z}) .

Therefore

f_{x x} = (f_{z} + f\overline{z})_{z} + (f_{z} + f\overline{z})\overline{z} = f_{z z} + 2f_{z \overline{z}} + f_{\overline{z} \overline{z}},

f_{y y} = ı (f_{z} − f\overline{z})_{y} = − (f_{z} − f\overline{z})_{z} + (f_{z} − f\overline{z})\overline{z} = −f_{z z} + 2f_{z \overline{z}} − f_{\overline{z} \overline{z}}

and thus

f_{x x} + f_{y y} = 4f_{z \overline{z}};

that is,

A = 0 = C and B = 4.

(d) Prove that if f is holomorphic, then

u_{x x} + u_{y y} = 0.

SOL’N:

u_{x x} + u_{y y} = ℜ (f_{x x} + f_{y y}) = 4ℜf_{\overline{z} z} = 0.

(e) Let u be the real part of a holomorphic function. Show that u_{z} is a holomorphic function. SOL’N:

u_{z \overline{z}} = (ℜf)_{\overline{z} z} = 0.

(f) Find necessary and sufficient conditions for the C^{1}-differential form f(z)dz + g(z)\overline{dz} to be closed. SOL’N: f(z)dz +g(z)\overline{dz} is closed iff d (f(z)dz + g(z)\overline{dz}) = 0. But

d (f(z)dz + g(z)\overline{dz}) = (∂ + \overline{∂}) (f(z)dz + g(z)\overline{dz}) = (−f\overline{z} + g_{z})dz\overline{dz}.

Thus f(z)dz + g(z)\overline{dz} is closed iff f\overline{z} = g_{z}. If one wants to use only real notation, then we rewrite

fdz + g\overline{dz} = (f + g)dx + ı(f − g)dy.

Thus the form is closed iff ı(f − g)_{x} = (f + g)_{y} (equivalent to the previous conclusion, of course).

3. (20 pts) State carefully (proofs \underline{not} required).

(a) Cauchy’s integral formula. SOL’N: If f is holomorphic in a domain D, γ is a closed path in D and a ∉ range γ, then

\frac{1}{2πı} ∫_{γ} \frac{dz}{f(z) − a} = I(γ, a)f(a).

(b) Liouville’s theorem.

SOL’N: A bounded entire function is constant.

(c) Morera’s theorem. SOL’N: If f(z)dz is a closed differential form on a domain D, then f is holomorphic in D.

(d) The maximum modulus principle. SOL’N: If the continuous function f on the closure of the domain D satisfies the mean value property in D and |f| has a relative maximum in D, then f is constant in the closure of D.

4. (10 pts) Let f be an entire function. Assume there exist an R > 0 and an M > 0 such that |f(z)| ≤ M|z|^{8.25} for all |z| > R. Prove that f is a polynomial. What is its degree? SOL’N: Let

f(z) = ∑_{n=0}^{∞} a_{n}z^{n}

be the Taylor series expansion for f. Cauchy’s estimate tells us that

|a_{n}| ≤ \frac{M|z|^{8.25}}{r^{n}} , for all r > R.

If n ≥ 9 and we let r → +∞, we conclude that

a_{n} = 0 for all n > 8

and thus f is a polynomial of degree at most 8.

5. (10 pts) Determine the order of vanishing at the origin of each of the following entire (NOTE: SOME OF THESE FUNCTIONS “BECOME” ENTIRE ONLY IF PROPERLY DEFINED AT THE ORIGIN) functions:

(a) (e^{z} − 1)^{3}, SOL’N: 3

(b) z sin z^{3}, SOL’N: 1 + 3 = 4

(c) z + z^{2} + z^{3}, SOL’N: 1

(d) \frac{sin z}{z}, and SOL’N: 1 − 1 = 0

(e) \frac{z^{2}−1}{z+1}. SOL’N: 0