The theory of functions of one complex variable with an emphasis on the theory of complex analytic (or holomorphic) functions of one complex variable. Typical topics include: Cauchy's integral formula, singularities, poles, meromorphic functions, Laurent and Taylor series, maximum modulus principle, ...
1
vote
0answers
20 views
radius of convergence of function defined by alternated addition and multiplication
Starting from $z\in\mathbb{C}$, define $r(z)$ starting from the initial value $v_{0}=0.0$ and repeat the following iteration from $1,2,\ldots$
$$v_{n+1} = \begin{cases} v_{n} + \frac{z^{n}}{n!} ...
0
votes
1answer
34 views
Does $f(z)=\sin\left(\dfrac{1}{z}\right)~\forall~z\in\mathbb C-\{0\}?$
Let $f$ be analytic in $\mathbb C-\{0\}$ such that $f\left(\dfrac{1}{n\pi}\right)=\sin\left(n\pi\right)~\forall ~n\in\mathbb Z.$
Does $f(z)=\sin\left(\dfrac{1}{z}\right)~\forall~z\in\mathbb ...
0
votes
0answers
30 views
What can we say about the entire functions that omit the value zero?
What can we say about the entire functions that omit the value zero?
My thought:
Since any nonconstant entire function comes arbitrarily close to any complex number the said functions are either ...
1
vote
1answer
39 views
Holomorphic function which is injective
Let $f$ be holomorphic in the upper half plane $\{\operatorname{Im} z> 0\}$. Suppose that $\operatorname{Im} f'(z)> 0$ for all $z$ in the half plane. Show that $f$ is 1-to-1.
1
vote
0answers
30 views
Zeta function zeros and analytic continuation
I'm learning about the zeta function and already discovered the intuitive proof of the Euler product and the Basel problem proof. I want to learn how to calculate the first zero of the Riemman Zeta ...
0
votes
2answers
36 views
Help with algebra involving residues
Evaluate $$\int_0^\infty \dfrac {\cos(mx)}{1+x^4}dx$$
I know I have to change the inegral to $-\infty$ as the lower limit. I understand the logic of the problem but it's the algebra in finding the ...
1
vote
2answers
42 views
Integration using residues
For the following problem from Brown and Churchill's Complex Variables, 8ed., section 84
Show that
$$ \int_0^\infty\frac{\cos(ax) - \cos(bx)}{x^2} \mathrm{d}x= \frac{\pi}{2}(b-a)$$
where $a$ and ...
-2
votes
1answer
49 views
Are functions differentiable on $\mathbb{C}$
What is the method i need to use to find out if functions are differentiable on $\mathbb{C}$
Some examples of past exam questions on this are..
(i) $f(x+iy)=x^2 -y^2+x+1+i(2xy+y)$
...
1
vote
2answers
42 views
Contour Integation $\int_\gamma \frac{\cos^2z}{z^2}$
I have the following question from a past exam paper that I'm not really sure how to evaluate. Any help would be appreciated...
Let $\gamma$ be the unit circle in $\mathbb{C}$ traversed in the ...
1
vote
0answers
36 views
f analytic on unit disk implies f is real on (-1,1)?
Looking at a problem in Wunsch. Suppose f is analytic on the unit disk. Then I am asked to show that f must be real on the interval (-1,1). If f = u + i v, then how do I show that v(x,0) = 0 for ...
0
votes
1answer
38 views
Find the set of all entire functions $f$ such that $|f(z)| \le |z|^{5/2}+|z|^{9/2}$ for $z \in D$ .
Find the set of all entire functions $f$ such that $|f(z)| \le |z|^{5/2}+|z|^{9/2}$ for $z \in D$ .
here clearly we can say$f(0)=0$ and it is bounded. so it must be the zero function. am I right?
2
votes
1answer
32 views
whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions
In each of the case below, determine whether or not there exist a non-constant entire function $f(z)$ satisfying the following conditions.
($1$) $f(0)=e^{i\alpha}$ and $|f(z)|=1/2$ for all $z \in Bdr ...
1
vote
0answers
32 views
Complex Analysis and Proper Holomorphic Maps
Prove there is no proper holomorphic map from the unit disc into the complex plane.
I know as $z$ approaches the boundary of the unit disc, $f(z)$ will approach infinity, hence unbounded.
Can we ...
0
votes
1answer
30 views
How can two functions be compared?
If the derivatives of all orders for two functions agree at one point how can two functions be compared?
There's no mention about the domain of those two functions. Can I suppose them to be ...
0
votes
0answers
22 views
How to determine where a function is complex differentiable
I know the definition of complex differentiability and also am aware that $f$ is complex differentiable at $z_0$ iff it is real differentiable at $z_0$ and that the partial derivatives satisfy the ...
