Math Problems Vault

Problem 5.1.1

True-False. If the assertion is true, quote a relevant theorem or reason; if false, give a counterexample or other justification.

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(a)

If $f(z) = u + iv$ is continuous at $z = 0$ , and the partials $u_x, u_y, v_x, v_y$ exist at $z = 0$ with $u_x = v_y$ and $u_y = -v_x$ at $z = 0$ , then $f'(0)$ exists.

(b)

If $f(z)$ is analytic in $\Omega$ and has infinitely many zeros in $\Omega$ , then $f \equiv 0$ .

(c)

If $f$ and $g$ are analytic in $\Omega$ and $f(z) \cdot g(z) \equiv 0$ in $\Omega$ , then either $f \equiv 0$ or $g \equiv 0$ .

(d)

If $f(z)$ is analytic in $\Omega = \left\{ z; \mathrm{Re}\left(z\right) > 0\right\}$ , continuous on $\overline{\Omega }$ with $\left|f(iy)\right| \leq 1$ ( $-\infty < y < +\infty$ ), then $\left|f(z)\right| \leq 1$ ( $z \in \Omega$ ).

(e)

If $\sum a_n z^n$ has radius of convergence exactly $R$ , then $\sum n^3 a_n z^n$ has radius of convergence exactly $R$ .

(f)

$\sin \sqrt{z}$ is an entire function.

Problem 5.1.2

(a)(b)(c)

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(a)

Let $f(z)$ be a complex-valued function of a complex variable. If both $f(z)$ and $zf(z)$ are harmonic in a domain $\Omega$ , prove that $f$ is analytic there.

(b)

(c)

Determine all entire functions $f$ such that $\left|f'(z)\right| < \left|f(z)\right|$ .

Problem 5.1.3

(a)(b)

Let $G$ be a region in $\mathbb {C}$ and suppose $u : G \to \mathbb {R}$ is a harmonic function.

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(a)

Show that $\dfrac {\partial u}{\partial x} - i \dfrac {\partial u}{\partial y}$ is an analytic function on $G$ .

(b)

Show that $u$ has a harmonic conjugate on $G$ if and only if $\dfrac {\partial u}{\partial x} - i \dfrac {\partial u}{\partial y}$ has a primitive (anti-derivative) on $G$ .

Problem 5.1.4

Suppose that $u$ and $v$ are real valued harmonic functions on a domain $\Omega$ such that $u$ and $v$ satisfy the Cauchy-Riemann equations on a subset $S$ of $\Omega$ which has a limit point in $\Omega$ . Prove that $u + iv$ must be analytic on $\Omega$ .

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Problem 5.1.5

Let $Q = [0,1] \times [0,1] \subset \mathbb {C}$ be the unit square, and let $f$ be holomorphic in a neighborhood of $Q$ . Suppose that

f(z + 1) - f(z) \text{ is real and} \geq 0 \quad \text{for } z \in [0, i].

f(z + i) - f(z) \text{ is real and} \geq 0 \quad \text{for } z \in [0, 1].

Show that $f$ is constant.

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Problem 5.1.6

Let $f$ be continuous on the closure $\overline{S}$ of the unit square

S = \left\{ z = x + iy \in \mathbb {C} : 0 < x < 1, 0 < y < 1\right\} ,

and let $f$ be analytic on $S$ . If $\mathrm{Re}\left(f\right) = 0$ on $\overline{S} \cap (\left\{ y = 0\right\} \cup \left\{ y = 1\right\} )$ , and if $\mathrm{Im}\left(f\right) = 0$ on $\overline{S} \cap (\left\{ x = 0\right\} \cup \left\{ x = 1\right\} )$ , prove that $f \equiv 0$ everywhere on $\overline{S}$ .

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Problem 5.1.7

(a)(b)

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(a)

Find the constant $c$ such that the function

f(z) = \frac{1}{z^4 + z^3 + z^2 + z - 4} - \frac{c}{z-1}

is holomorphic in a neighborhood of $z = 1$ .

(b)

Show that the function $f$ is holomorphic on an open set containing the closed disk $\left\{ z : \left|z\right| \leq 1\right\}$ .

Problem 5.1.8

(a)(b)(c)

Let $P(z)$ be a polynomial of degree $d$ with simple roots $z_1, z_2, \ldots , z_d$ . A ``partial fractions'' expression of $\dfrac {1}{P}$ has the form:

\frac{1}{P(z)} = \sum _{n=1}^{d} \frac{c_n}{z - z_n}. \tag {*}

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(a)

Give a direct formula for $c_n$ in terms of $P$ .

(b)

Show that $\dfrac {1}{P(z)}$ really has a representation of the form (*).

(c)

Give a formula similar to (*) that works when $z_1 = z_2$ but all other roots are simple.

Problem 5.1.9

Suppose $f$ is meromorphic in a neighborhood of $\overline{D}$ ( $D = \left\{ \left|z\right| < 1\right\}$ ) whose only pole is a simple one at $z = a \in D$ . If $f(\partial D) \subseteq \mathbb {R}$ , show that there is a complex constant $A$ and a real constant $B$ such that

f(z) = \frac{Az^2 + Bz + \overline{A}}{(z-a)(1-\overline{a}z)}.

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Problem 5.1.10

(a)(b)

Let $K_1, K_2, \ldots , K_n$ be pairwise disjoint disks in $\mathbb {C}$ , and let $f$ be an analytic function in $\mathbb {C} \setminus \bigcup_{j=1}^{n} K_j$ . Show that there exist functions $f_1, f_2, \ldots , f_n$ such that

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(a)

$f_j$ is analytic in $\mathbb {C} \setminus K_j$ , and

(b)

$f(z) = \sum_{j=1}^{n} f_j(z)$ for $z \in \mathbb {C} \setminus \bigcup_{j=1}^{n} K_j$ .

Problem 5.1.11

Recall that a divisor $D_f$ of a rational function $f(z)$ on $\mathbb {C}$ is a set $\left\{ p \in \mathbb {C} \cup \left\{ \infty \right\} \right\}$ , consisting of zeros and poles $p$ of $f(z)$ (including the point $\infty$ ), counted with their multiplicities $n_p \in \mathbb {Z}$ . Let $f$ and $g$ be two rational functions with disjoint divisors. Prove that

\prod _{p \in D_f} g(p)^{n_p} = \prod _{q \in D_g} f(q)^{n_q}.

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Problem 5.1.12

(a)(b)(c)

Let $f(z)$ be the ``branch'' of $\log z$ defined off the negative real axis so that $f(1) = 0$ .

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(a)

Find the Taylor polynomial of $f$ of degree $2$ at $-4+3i$ , simplifying the coefficients.

(b)

Find the radius of convergence $R$ of the Taylor series $T(z)$ of $f(z)$ at $-4+3i$ .

(c)

Identify on a picture any points $z$ where $T(z)$ converges but $T(z) \neq f(z)$ , and describe the relationship between $f$ and $T$ at such points. If there are no such points, is this something special to this example, or a general impossibility? Explain and/or give examples.

Problem 5.1.13

Let $f$ be the analytic function defined in the disk $\Delta = \left\{ z : \left|z-4\right| < 4\right\}$ so that $f(z) = z^{1/3}(z+1)^{1/2}$ in $\Delta$ and $f(x)$ is positive for $0 < x < 8$ . An analytic function $g$ in $\Delta$ is obtained from $f$ by analytic continuation along the path starting and ending at $z = 4$ (see Fig. 5.2). Express $g$ in terms of $f$ .

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Problem 5.1.14

(a)(b)(c)

Define

f(z) = \frac{e^{\sqrt z} - e^{-\sqrt z}}{\sin \sqrt z}.

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(a)

Where is $f$ single-valued and analytic?

(b)

Classify the singularities of $f$ .

(c)

Evaluate $\int_{\left|z\right|=25} f(z)\, dz$ .

Problem 5.1.15

(a)(b)

Let $\Omega$ be the plane with the segment $\left\{ -1 \leq x \leq 1, y = 0\right\}$ deleted. For which of the multi-valued functions

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(a)

$f(z) = \dfrac {z}{\sqrt{1-z^2}}$ ,

(b)

$g(z) = \dfrac {1}{\sqrt{1-z^2}}$ ,

can we choose single-valued branches which are holomorphic in $\Omega$ . Which of these branches are (is) the derivative of a single-valued holomorphic function in $\Omega$ . Why?

Problem 5.1.16

(a)(b)

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(a)

Let $D \subset \mathbb {C}$ be the complement of the simply connected closed set $\left\{ e^{\theta +i\theta } : \theta \in \mathbb {R}\right\} \cup \left\{ 0\right\}$ . Let $\log$ be a branch of the logarithm on $D$ such that $\log e = 1$ . Find $\log e^{15}$ . Justify your answer.

(b)

Let $\gamma$ denote the unit circle, oriented counterclockwise. By lifting the integration to an appropriate covering space, give a precise meaning to the integral $\int_\gamma (\log z)^2\, dz$ and find all possible values which can be assigned to it.

Problem 5.1.17

Find the most general harmonic function of the form $f(\left|z\right|)$ , $z \in \mathbb {C} \setminus 0$ . Which of these $f(\left|z\right|)$ have a single valued harmonic conjugate?

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Problem 5.1.18

Consider the regular pentagram centered at the origin in the complex plane. Let $u$ be the harmonic function in the interior of the pentagram which has boundary values $1$ on the two segments shown and $0$ on the rest of the boundary. What is the value of $u$ at the origin? Justify your claim.

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Problem 5.1.19

Suppose $G$ is a region in $\mathbb {C}$ , $[0,1] \subset G$ , and $h : G \to \mathbb {R}$ is continuous. $h|_{G \setminus [0,1]}$ is harmonic, does this imply that $h$ is harmonic on $G$ ?

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Problem 5.1.20

Let $\gamma$ be an arc of the unit circle. Suppose that $u$ and $v$ are harmonic in $D = \left\{ z : \left|z\right| < 1\right\}$ and continuously differentiable on $D \cup \gamma$ . If the boundary values satisfy $u = v$ on $\gamma$ and the radial derivatives satisfy $\dfrac {\partial u}{\partial r} = \dfrac {\partial v}{\partial r}$ on $\gamma$ , prove that $u = v$ in $D$ .

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Problem 5.1.21

Use conformal mapping to find a harmonic function $U(z)$ defined on the unit disc $\left\{ \left|z\right| < 1\right\}$ such that

\lim _{r \to 1^-} U(re^{i\theta }) = \begin{cases} +1, & \text{for } 0 < \theta < \pi , \\ -1, & \text{for } \pi < \theta < 2\pi . \end{cases}

Give the correct determination of any multiple-valued functions appearing in your answer.

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Problem 5.1.22

Determine all continuous functions on $\left\{ z \in \mathbb {C} : 0 \leq \left|z\right| \leq 1\right\}$ which are harmonic on $\left\{ z : 0 < \left|z\right| < 1\right\}$ and which are identically $0$ on $\left\{ z \in \mathbb {C} : \left|z\right| = 1\right\}$ .

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Problem 5.1.23

(a)(b)(c)(d)

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(a)

\frac{1}{2\pi } \int _0^{2\pi } \log \left|f(re^{i\theta })\right|\, d\theta = \sum _{j=1}^{n} \log \frac{r}{\left|a_j\right|}.

Prove this.

(b)

With the hypotheses and notations of (a), let $n(t)$ be the number of $a_j$ ( $j = 1, 2, \ldots , n$ ) such that $\left|a_j\right| < t$ . Using Jensen's formula, show that

\int _0^r \frac{n(t)}{t}\, dt = \frac{1}{2\pi } \int _0^{2\pi } \log \left|f(re^{i\theta })\right|\, d\theta .

(c)

For $r < R$ deduce an estimate on $n(r)$ in terms of $\max_{0 \leq \theta \leq 2\pi } \log \left|f(Re^{i\theta })\right|$ .

(d)

What can be said about the zeros of bounded holomorphic functions in the unit disc?

Problem 5.1.24

Let $B(z) = \displaystyle \prod_{k=1}^{n} \dfrac {a_k - z}{1 - \overline{a_k} z}$ , where $a_1, a_2, \ldots , a_n$ are distinct points in $D \setminus \left\{ 0\right\}$ . Show that

B(z) = \prod _{k=1}^{n} \frac{1}{\overline{a_k}} + \sum _{k=1}^{n} \frac{1}{\overline{a_k} \, \overline{B'(a_k)} (1 - \overline{a_k} z)}.

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Problem 5.1.25

Does there exist an analytic function in $\left|z\right| < 1$ such that $0 < \left|f(\frac{1}{n})\right| < e^{-n}$ for $n = 2, 3, 4, \ldots$ ? Justify your answer.

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Problem 5.1.26

Let $D = \left\{ z \in \mathbb {C} : \left|z\right| < 1\right\}$ . Find all holomorphic functions $f : D \to \mathbb {C}$ such that $f\! \left(\frac{1}{n} + ie^{-n}\right)$ is real for all integers $n \geq 2$ .

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