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, ...

learn more… | top users | synonyms (2)

2
votes
1answer
205 views

Riemann surface diagram of $\frac1{\sqrt z}$

I wish to find what the Riemann surface diagram of $\dfrac1{\sqrt z}$ looks like. The problem I'm having is that this function is not defined for $z=0$, and that this function doesn't map any ...
5
votes
1answer
737 views

Use rectangular contour to integrate $\sin(ax)/(\exp(2\pi x)-1)$

I have been self-studying CA and find it very interesting. So, working through problems in a book I have, I ran across $$\int_{0}^{\infty}\frac{\sin(ax)}{e^{2\pi ...
7
votes
1answer
556 views

The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $

This is an exercise from Remmert. The power series $\sum\limits_{n=1}^{\infty} \frac {z^{n} }{ n^{2}} \ $ has radius of convergence $1 \ $. Show that the function it represents is injective in $\{ z ...
0
votes
1answer
255 views

Holomorphic functions

Let $f:\mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function such that for an open interval $V \subset \mathbb{R}$ the following holds: $f(V)=0$. Does there exist an open set $U \subset ...
1
vote
1answer
315 views

To show given function is smooth

Let $\phi$ be upper semi-continuous function defined on $\Omega\subset \mathbb C^2$. Let $\Omega_n= \{z \in \Omega: d(z,\partial \Omega) > \frac1n \}$. Let $\chi\in C_c^\infty$ of $|z_1|$, ...
1
vote
1answer
172 views

Existence of the absolute value of the limit implies that either $f \ $ or $\bar{f} \ $ is complex-differentiable

This is an exercise from Remmert. Let D be domain in $\mathbb{C} \ $, and $f : D \rightarrow \mathbb{C} \ $ a real-differentiable function. Suppose that for some $ c \in D $ the limit $\lim_{h \to ...
2
votes
0answers
137 views

compute this integration

$$\int_{x_1}^{x_2}\frac{\sqrt{\frac{1}{3}x^3+a}}{(1-x)\sqrt{x}\sqrt{-\frac{4}{3}x^3+x^2-a}}dx$$ where $a\in(0,\frac{1}{12})$ is a constant. In this case, $-\frac{4}{3}x^3+x^2-a=0$ has exactly two ...
4
votes
2answers
342 views

Fourier transform on $\mathbb{R}^3$

I've been stuck on this one homework problem for nearly a day now, so I'd be really thankful for any pointers. The problem is to find $$\int\int\int_{\mathbb{R}^3} \frac{1}{\mathbf{x}^2+1}e^{-2\pi i ...
1
vote
1answer
100 views

$\int_{\mathbb{C}} \frac{\partial f }{ \partial z} dxdy = \int_{\mathbb{C}} \frac{\partial f }{ \partial \bar{z}} dxdy = 0$

f is a complex function, $C^{1} $ on $ \mathbb{C} \ $, $ f = 0 $ on $ K^{c} \ $, where $K$ is a compact. Then $\int_{\mathbb{C}} \frac{\partial f }{ \partial z} dxdy = \int_{\mathbb{C}} ...
7
votes
1answer
475 views

Physical interpretation of the generating function for the Bessel functions.

It is well known that the generating function for the Bessel function is $$f(z) = \exp \left (\frac12 \left (z - \frac1z \right ) w \right ).$$ So, we have $$f(z) = \sum_{\nu = -\infty}^{\infty} ...
3
votes
1answer
277 views

continuous function from $[0,1]$ to $\mathbb{C}$

Let $T=\{z\in \mathbb{C} :|z|=1\}$ and $f:[0,1]\rightarrow \mathbb{C}$ be continuous with $f(0)=0$ and $f(1)=2$. I need to show that there exists at least one $t_0\in [0,1]$ such that $f(t_0)\in T$. ...
4
votes
2answers
160 views

Showing $f(\zeta)=\frac{1}{\pi}\int_{|z|<1}\frac{f(z)dxdy}{(1-\bar{z}\zeta)^2}$

I'm doing practice physics qualifying exam problems and came across this one I didn't know how to solve: Show that if $f(x)$ is bounded and analytic for $|z|=|x+iy|<1$, then ...
1
vote
1answer
112 views

Find the saddle points $z_{1},z_{2}$ of $f(z)=\frac{(z - 1)^{2}(z + 1)}{z^{2}}$

Find the saddle points $z_{1},z_{2}$ of $f(z)=\frac{(z - 1)^{2}(z + 1)}{z^{2}}$ Does anyone can help me with this problem? $z_{0}$ is a saddle point of an analytic function f if and only if ...
4
votes
2answers
1k views

Computing $\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}dx$ using residue calculus

I need to find $\displaystyle\int_{-\infty}^{\infty} \frac{\cos x}{x^{2} + a^{2}}\ dx$ where $a > 0$. To do this, I set $f(z) = \displaystyle\frac{\cos z}{z^{2} + a^{2}}$ and integrate along the ...
2
votes
1answer
181 views

Show Schwarz-Christoffel retains the same form

Set $w=\phi(z)=i\frac{1+z}{1-z}$ (which maps the unit disk in the complex plane to the upper half of the complex plane). Show that the Schwarz Christoffel formula, $$f(z)=A_1\int_0^z \frac ...
2
votes
0answers
225 views

Schwarz-Christoffel Complex Mapping

Verify that the Schwarz-Christoffel mapping of $\mathbb H$ onto the infinite half strip described by $|\Re(z)|<\frac \pi 2$ and $\Im(z)>0$ is given by the arcsine function. What does that ...
1
vote
0answers
101 views

Continuous but not uniformly continuous in $\text{GL}(2,\Bbb{C})$

Essentially, I'm trying to find an example of a function that is continuous but not uniformly continuous on $\text{GL}(2,\Bbb{C})$. I'm aware that this group is isomorphic (up to constant multiples) ...
8
votes
2answers
303 views

Prove that $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's identity.

I'm trying to derive the functional equation $2^{2z-1}\Gamma(z)\,\Gamma(z+\frac{1}{2})=\sqrt{\pi}\,\Gamma(2z)$ using Gauss's formula: ...
3
votes
0answers
143 views

Does $\displaystyle \lim_{m \to +\infty}f_{2,m}(x)$ converge?

This is related to a previous question where, as stated there, $f_{2}(n)$ gives the greatest power of $2$ that divides $n$. Specifically the sequence $\lbrace ...
4
votes
1answer
1k views

Finding closed forms for $\sum n z^{n}$ and $\sum n^{2} z^{n}$

Using the identity $\frac{1}{1-z} = 1 + z + z^2 + \ldots$ for $|z| < 1$, find closed forms for the sums $\sum n z^n$ and $\sum n^2 z^n$. My solution: Because $\displaystyle1 + z + z^{2} + ...
1
vote
1answer
360 views

Relative maximum and minimum of the modulus of a function

Let $$f(z)=(z -1)(z -4)^{2}$$ Find the lines (through $z=2$) on which $|f(z)|$ has a relative maximum, and the ones on which $|f(z)|$ has a relative minimum. MY ATTEMPT: "The line z= 2" is the line ...
3
votes
1answer
97 views

Prove that $\sin(z_{1}+z_{2})=\sin z_{1} \cos z_{2} + \cos z_{1} \sin z_{2}$

Prove that $\sin(z_{1}+z_{2})=\sin z_{1} \cos z_{2} + \cos z_{1} \sin z_{2}$ My solution Let $z_{2}$ be a fixed real number. Then, $f(z)=\sin(z + z_{2})$ and $g(z)=\sin z\cos z_{2}+\cos z\sin ...
1
vote
0answers
78 views

Show that any closed polygonal path can be decomposed into a finite union of simple closed polygonal paths

Show that any closed polygonal path can be decomposed into a finite union of simple closed polygonal paths and line segments traversed twice in opposite directions. MY SOLUTION Suppose ...
1
vote
1answer
494 views

Prove that every convex region is simply connected

Prove that every convex region is simply connected Could anyone help?
8
votes
2answers
464 views

entire function with only finitely many zeros

I saw the following exercise: If $f:\mathbb{C}\rightarrow\mathbb{C}$ is an entire, non-constant function with only finitely many zeros, then either $|f(z)|\rightarrow \infty$ for ...
8
votes
1answer
489 views

Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$

Show that if f is analytic in $|z|\leq 1$, there must be some positive integer n such that $f(\frac{1}{n})\neq \frac{1}{n+1}$ MY SOLUTION If $f(\frac{1}{n})=\frac{1}{n+1}$, then for all points ...
4
votes
1answer
1k views

Find a power series expansion for $\frac{1}{z}$ around $z = 1 + i.$

Find a power series expansion for $\frac{1}{z}$ around $z = 1 + i.$ My Solution For any complex $\alpha$,$\frac{1}{z}=\frac{1}{\alpha+z-\alpha}=\frac{1}{\alpha[1+\frac{z-\alpha}{\alpha}]}$ ...
1
vote
1answer
53 views

Can we proof $B=C^*$ given $|B-Ae^{-j\omega\delta}|=|C-Ae^{+j\omega\delta}|$

I want to ask for verification about whether this equation can be proven. If so, what is the best way to approach it? I tried this way... but I don't know how to continue on. ...
2
votes
2answers
997 views

Complex conjugation continuous function

How can I show that complex conjugation is a continuous function? I tried looking at open sets $U$ and then the preimage. Can assume preimage is not empty so that if $z$ is in preimage then $f(z) \in ...
1
vote
1answer
164 views

Suppose a region S is simply connected and contains the circle

Suppose a region $S$ is simply connected and contains the circle $C =\{z:|z-\alpha|=r\}$. Show then that $S$ contains the entire disc $D=\{z:|z-\alpha|\leq r\}$. HINT OF THE BOOK: Show that since $S$ ...
1
vote
1answer
81 views

Finding a given path integral

Let $C$ be the path determined by the square with vertices $(1,1),(-1,1), (-1,-1), (1,-1)$, in the counterclockwise direction. How would one go about finding the following integral? $$\int_C ...
0
votes
2answers
990 views

Proof of the Schwarz Lemma

I have a question which is (the Schwarz Lemma): Suppose that $f:\mathbb{D}\rightarrow\mathbb{D}$ is holomorphic and suppose that $f(0)=0$, show that $\lvert f(z)\rvert \leq \lvert z \rvert ...
0
votes
1answer
95 views

How is this a harmonic conjugate when it is not harmonic itself?

Suppose $f(z) = z^2$ This function has the component functions $u(x,y) = x^2 - y^2$, $v(x,y) = 2xy$ And it says in a book I'm reading that $v$ is a harmonic conjugate of $u$. But v is not harmonic ...
4
votes
1answer
271 views

holomorphic function is real analytic?

$f$ is a holomorphic function on $\mathbb C^n$. If we regard $f$ as a function $F$ from $\mathbb R^{2n} \to \mathbb R^2$, is it necessarily that $F$ is real analytic?
1
vote
0answers
152 views

Complex Analysis: Conformal mapping, a strange question I encountered

I need to find a mapping for $z + \frac{1}{z}$ to the $x$-axis. I have that $f(i) = 0$, $f(-1)= -2$ and $f(1) = 2$. I am not sure as to what my professor seems to be asking us to think of in this ...
1
vote
1answer
259 views

Question about The Weierstrass Factorization Theorem

According to wikipedia, "...the Weierstrass factorization theorem in complex analysis, named after Karl Weierstrass, asserts that entire functions can be represented by a product involving ...
0
votes
3answers
134 views

Find an object $x$ such that $1=\sum_{n>0} n^n\cdot x^n$

$1=\sum_{n>0} (nx)^n$ Dont have any solution in $\mathbb C$? Are there other types of equations with no solutions in $\mathbb C$? Can one define an object that satisfyes these equations? Does it ...
4
votes
1answer
844 views

At which points is this function differentiable/analytic?

At which points (if any) is this function differentiable? At which points is it analytic? $f(x+iy) = x^2 + iy^2$ I applied the Cauchy Riemann equations and got the result that $y=-x$. So then am I ...
2
votes
4answers
938 views

Proving a function is constant

Let $f$ be an analytic function such that $f(z)$ is an element of $\mathbb R$ for all $z$ element of $\mathbb C$. Prove $f$ is constant. Here's what I have done - $f(z) = c + i0$, where $c$ is an ...
6
votes
3answers
563 views

“Unsolvable” Equations

I was just playing around with some equations the other day and losing interest I started writing down what were mostly "random" equations at first. But, I realized that there's something special ...
8
votes
1answer
536 views

sum of series involving coth using complex analysis

I am self-studying complex analysis, so I am a rookie. I ran across an interesting series I am trying to evaluate using CA. Show that $$\sum_{n=1}^{\infty}\frac{\coth(\pi ...
0
votes
1answer
211 views

Let $f(z)$ be analytic and nonzero in a region R. Show that $|f(z)|$ has a minimum value in R that occurs on the boundary.

Let $f(z)$ be analytic and nonzero in a region R. Show that $|f(z)|$ has a minimum value in R that occurs on the boundary. I think you should use the Maximum-Modulus Theorem for the function $1/f(z)$ ...
3
votes
2answers
449 views

How does a conformal mapping preserve angles in hyperbolic geometry?

Suppose I have a sector $D = \{0 < \arg z < \alpha\}$ where $\alpha \leq 2\pi$. If I apply the function $w = \frac{\zeta - i}{\zeta + i}$ from the upper half plane to the unit disc ($\zeta = ...
4
votes
3answers
887 views

Are there any simple ways to see that $e^z-z=0$ has infinitely many solutions?

Joseph Bak and Donald Newman's complex analysis book (p.236) has a proof that the equation $e^z-z=0$ has infinitely many complex solutions: I'm curious if there are any particularly elegant ways ...
2
votes
1answer
51 views

How can $\int_{|z|=2}\frac{z^4}{z^5-z-1}dz$ be computed?

I'm going through some practice prelim exams, and one of the questions asks to compute $$ \int_{|z|=2}\frac{z^4}{z^5-z-1}dz. $$ The integrand is not quite in the form $f'/f$ to count zeroes and poles ...
2
votes
1answer
359 views

Help making a proof about conformal mappings rigorous

$\newcommand{\Rea}{{\text{Re}}} \newcommand{\Ima}{{\text{Im}}}$ Here is the question straight of the book: http://bit.ly/JtbjsW Prove the following: Suppose $f: \Omega \rightarrow \mathbb{C}$ is ...
0
votes
1answer
98 views

Why in Complex Variable they have to use the word 'Vanish' [closed]

I don't understand in what circumstance something could vanish when doing the complex integration. Vanish this word sounds too abstract to me.
5
votes
1answer
262 views

An entire function $g$ such that $|g(z^2)| \leq e^{|z|}$ and $g(m) = 0 \quad \forall m \in \mathbb{Z}$ is identically $0$

I have been trying to solve the following exercise from a collection of old complex analysis qualifier exams. Suppose that $g$ is an entire function that satisfies the inequality $|g(z^2)| \leq ...
4
votes
3answers
320 views

Riemann sphere and Maps

Could somebody please clarify the following for me? I am not too clear about the relationship between the Riemann sphere and Möbius maps. I know that we can through projection make some Möbius maps ...
9
votes
1answer
200 views

Existence of an entire function with algebraically independent derivatives

Let $\mathbb{A}$ be the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. A collection of functions $F=\lbrace f_i:X \rightarrow\mathbb{C}\rbrace$ is said to be algebraically independent over ...