I am studying for my (introductory) complex analysis final exam tomorrow. I am practicing an old final exam, which unfortunately has no answer key. Here is a link:
http://www.math.ubc.ca/Ugrad/pastExams/Math_300_December_2008.pdf
I just want to know if I am on the right track for Problem 1, which is collection of 10 (kind of tricky!) true/false questions. Here are my attempts:
1) If $f(z)$ satisfies Cauchy-Riemann equations at $z_0$, then $f(z)$ is differentiable at $z_0$.
False. For differentiability, one also needs continuity of partial derivatives.
2) If $f(z)$ has a pole at $z_0$, then $\lim_{z\to z_0}|f(z)|=\infty$.
True. Though I am not sure if I know how to prove it rigorously.
3) If $f(z)$ is analytic in a domain $D$ containing a simple closed contour $\Gamma$, then $\int f(z)dz=0$.
False. For conclusion to be true in general, one needs simply-connected domain.
4) If the two power series $\sum_{k=0}^\infty a_k (z-z_0)^k$ and $\sum_{k=0}^\infty b_k (z-z_0)^k$ converge to the same function in the disk $\{|z-z_0|\}$, then $a_k=b_k$ for all $k$.
I think the answer is True for this one. I believe this follows from Taylor theorem. The coefficients $a_k$ and $b_k$ are determined from derivatives of $f$, so they must be equal. Is this correct?
5) There does not exist any function $f(z)$ which is analytic at the point $0$ and nonanalytic everywhere else.
I think the answer is False. I can't think of a counterexample. Maybe the function $|z|^2$ ?
6) The function $\text{Log}(z^2)$ is analytic for all values of $z$ except those on the negative real axis.
False. The function $\text{Log}(z^2)$ is analytic for all values of $z$ except those on upper half of imaginary axis.
7) Any entire function is the complex derivative of another entire function
True. This follows from Cauchy Integral Formula.
8) If $f(z)$ has an essential singularity at $z_0$, then Res$((z-z_0)f(z); z_0)=0$.
I am pretty sure this is false. For example, $f(z)=e^{1/z}$ would be a counter-example.
9) If $f(z)$ and $g(z)$ have a simple poles at $0$, then $(fg)(z)$ has a simple pole at $0$.
I think it is true. By the way, I am not sure if the notation above stands for product, or composition. The question doesn't indicate the usage. But I think in either case, the statement would be true. Any comments on this one?
10) If the disk of convergence of the Taylor series of a function $f(z)$ is $\{|z|=2\}$, then the disk of convergence for the Taylor series of $f(z^2)$ is $\{|z|=4\}$.
Frankly, I am completely struck at this one. I have not the slightest idea of relating radii of convergence for the functions $f(z)$ and $f(z^2)$.
Any help and feedback on any of the questions is much appreciated. I suspect I have made couple mistakes above, apart from the ones I am stuck on. Confirming one of the my answers as correct would be just as awesome! :)
Thanks.
For 2) Lets say that $z_0$ is a pole of order $n \geq 1$. Then there exists an analytic function $g(z)$ with $g(z_0)\neq 0$ so that $f(z)=\frac{g(z)}{(z-z_0)^n} $. From here it is easy.
For 5): The definition of analiticity is that the power series must converge in a neighborhood of $0$. But then $f(z)$ is analityc in that neighborhood.
Note that $f(z)=|z|^2$ is differentiable at $z=0$ but not analytic...
For 9) That is the product, and you can actually show that $fg(z)$ has a pole of order $2$.
For 10)
Let $f(z)=\sum_{n=0}^\infty a_nz^n \,.$ Then $f(z^2)=\sum_{n=0}^\infty a_nz^{2n}$. It is easy to prove that the first series converges for all $|z| \leq a$ if and only if the second series converges for all $|z| \leq \sqrt{a}$.