Complex Analysis in Several Variables, MTH 3108

Home Assignement 1

Professor M.Shubin

Fall 1997

Textbook:

Introduction to Complex Analysis in Several Variables, by Lars Hörmander. 3rd edition, North Holland, 1990

1. Check whether the following functions are subharmonic or not:

• (a) f(z)=|z| in $\hbox{{\bbb C}}$;

• (b) f(z)=-|z| in $\hbox{{\bbb C}}$;

• (c) $f(z)=\log \vert z\vert$ in $\hbox{{\bbb C}}$;

• (d) $f(z)=-\log \vert z\vert$ in $\hbox{{\bbb C}}$;

• (e) $f(z)=-\log \vert z\vert$ in $\hbox{{\bbb C}}\setminus\{0\}$.

2. (a) Let u=u(z) be analytic in $D\setminus \{0\}$ where D is the unit disc $\{z:\;\vert z\vert<1\}$, and

$$\lim_{z\to 0} zu(z)=0\;.\leqno (1)$$

Use the Cauchy formula to prove that the singularity at is removable, i.e. there exists a function $\hat u$ which is analytic in D and coincides with u on $D\setminus \{0\}$.

(b) Instead of (1) assume that there exists an integer N>0 such that

$$\lim_{z\to 0} z^Nu(z)=0\;.\leqno (2)$$

Prove that u can be extended to a meromorphic function in D.

3. Use the Cauchy formula to prove the Liouville theorem: if u is an entire function (i.e. a function which is analytic in $\hbox{{\bbb C}}$) and there exist constants C,N>0 such that

$$\vert u(z)\vert\le C(1+\vert z\vert)^N,\ z\in\hbox{{\bbb C}},$$

then u is a polynomial in z.

4. Find all solutions in $\hbox{{\bbb C}}$ for the equation

$${\partial u\over \partial\bar{z}}=e^z\;.$$

5. (a) Prove that any harmonic function u in $\hbox{{\bbb C}}$ can be presented in the form

$$u(z)=f(z)+g(\bar{z})\;,\leqno (3)$$

where f,g are entire functions (i.e. functions which are analytic in $\hbox{{\bbb C}}$).

(b) Is the presentation (3) unique? If not, describe all possible presentations of u in this form.