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)

0
votes
0answers
13 views

Problem with statement of Needham's Visual Complex analysis exercise 15 chapter 3 page 185

The exercise has a diagram with 3 clockwise points q, r, s on a citcle and a 4th point z on the circle between q and s, and a second diagram with the same three points q, r, s but with z on the circle ...
2
votes
6answers
53 views

Computing $\int_{0}^{\infty} \frac{x^2}{(x^2 + a^2)^2}dx$

I want to compute $$\int_{0}^{\infty} \frac{x^2}{(x^2 + a^2)^2}dx$$ and have tried applying trig substution with $x = a\tan(t)$, but things get a bit messy at the very end. I get $$ \left [ ...
2
votes
3answers
54 views

Computing $\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx$

I wish to compute $$\int_{0}^{\infty} \frac{x^2 + a}{x^6 + a^3}dx, \quad a>0$$ but have no contour to work with. Does anyone have ideas on how to compute this integral?
0
votes
0answers
25 views

Injectivity of analytic $f: D(0, r) \mapsto \mathbb{C}$ in the range $D(0, \min\{|f(z)|: |z| = r\})$

If $f$ is analytic on $\{|z| \leq r\}$ and $f(0) = 0, f'(z) \neq 0$. Let $\rho = \min\{|f(z)|: |z|=r\}$. Is it true that if $|w| \leq \rho$ in the image of $f$, then $f(z) = w$ has only one zero on ...
2
votes
2answers
16 views

Laurent Series of $(z^2 + 3z + 2)e^{\frac{1}{z+1}}$

I want to find the Laurent series of $(z^2 + 3z + 2)e^{\frac{1}{z+1}}$ around $z_0 = -1$. However, since this is not a fraction in the form $\frac{a}{z-b}$, I am not sure how to calculate it.
2
votes
1answer
23 views

Generalised Cauchy Integral formula

Is the step I have highlighted in the proof incorrect? We only know $f$ is holomorphic for $|z|<1$, so why are we able to use the generalised Cauchy's formula on the curve |$z|=1$?
0
votes
0answers
19 views

Conformal isomorphisms from disc to half-disc fixing 1 and -1

I am preparing for a complex analysis qual and ran into this problem on an old exam. Find all conformal isomorphisms from the unit disk $\mathbf{D}=\{z\in\mathbf{C} : |z| < 1\}$ to the semi-disk ...
1
vote
1answer
26 views

Finite number of zeros if $f$ is analytic and satisfies homologous condition

I was reading Conway's complex analysis and I encounter the following exercise. I appreciate if someone can help me. Let $G$ be open and suppose that $\gamma$ is closed rectifiable curve in $G$ ...
2
votes
4answers
119 views

Complex number, series

Show that $$\frac{1}{z^2}=1+\sum_{n=1}^\infty (n+1)(z+1)^n$$ when $|z+1|<1$ I'm having problems to resolve this type of exercise since my book has virtually no exercises of this type, these ...
1
vote
1answer
20 views

Solve polynomial equation in $\mathbb{C}[x]$

Find the polynomials $f,g \in \mathbb{C}[x]$ with complex coefficients such that: $$f(f(x))-g(g(x))=1+i,\\f(g(x))-g(f(x))=1-i$$ for all $x\in\mathbb{C}$. I think I have this problem almost ...
1
vote
1answer
26 views

Complex number, series representation

Show that for any finite value of $z$ $$e^z=e+e\sum_{n=1}^\infty \frac{(z-1)^n}{n!}$$ For $z=1$ $$f(z)=f(z_0)+\sum f^{(n)}(z_0)\frac{(z-z_0)^n}{n!}$$ equality is checked, but I do not know how to ...
1
vote
1answer
25 views

Compute $\frac{f(i\frac{\pi}{2})}{f(i\pi)}$ for an analytic function.

I came across this problem and I'm having a little trouble. Let $f$ : $\mathbb{C} \rightarrow \mathbb{C}$ be a nonconstant analytic function. Assume that on $\mathbb{R}$ the function is real-valued ...
-1
votes
0answers
34 views

Explaining Cauchy's Theorem

I need to explain why the cauchy theorem does not apply to this integral $$\int_\gamma \bar{z}\ dz,$$ where $\gamma$ is the segment straight line [0,2+i] I found that the integral equals $5/2$. I ...
-1
votes
0answers
20 views

Verifying a Möbius transformation

This is the formula given for a transformation that sends (z2, z3, z4) to (1, 0, $inf$).. http://imgur.com/BmNpaiU 2 I reversed z2 and z3 to keep it in this form, but had no luck at all. I also ...
0
votes
1answer
29 views

Applying Cauchy's theorem

Why is the part highlighted in green equal to zero?
0
votes
0answers
22 views

Prove $SU(2)$ is isomorphic to the group of quaternions of norm 1

How could I start finding the isomorphism? Intuitively, a quaternion can be expressed as two complex numbers $a+bi+cj+dk=a+bi+(c+di)j$, and as an element of $SU(2)$ is $\left[ \begin{array}{ c c } ...
0
votes
1answer
14 views

Complex Vector spaces inner product superposition axiom

In my studies of Quantum mechanics, the following problem with complex vector spaces has come up, specifically as regards the inner product in such a space. Now in Shankars "Principles of Quantum ...
0
votes
0answers
13 views

Complex Differentiability and its difference to real differentiability

I am currently studying a course on complex analysis and complex differentiability is defined as: $f : U \rightarrow \mathbb{C}$, where $U$ is a domain, is complex differentiable if and only if it is ...
11
votes
3answers
154 views

Does $\sqrt{i + \sqrt{i+ \sqrt{i + \sqrt{i + \cdots}}}}$ have a closed form?

I've been brushing up on my complex analysis recently, and I've come across a problem that's stumped me: What are the real and imaginary parts of $$\sqrt{i+\sqrt{i+\sqrt{i+\sqrt{i+\cdots}}}} ?$$ I ...
0
votes
1answer
40 views

Analytic inverse of $f(z) \neq 0, f(0) = 0, f'(z) \neq 0 $ within minimum modulus on boundary.

Suppose $f(z)$ is analytic on closed disk of radius $r$ and $f(0)=0$, $f'(z) \neq 0$. Show that $f$ has an analytic inverse on $\{|z| \leq m\}$ where $m$ is the minimum of $|f(z)|$ on $\{|z| = r\}$. ...
-2
votes
0answers
29 views

Question about a challenging problem in complex analysis, any ideas are welcome [on hold]

Is there any body who have any idea to solve this problem? It seems that it is a challenging problem. The problem is as follows:
3
votes
0answers
31 views

True or False: If f is analytic and maps the deleted unit disk to the unit disk, then 0 is not a pole for f.

I am studying for my final exam in complex variables and I ran across this true or false question. True or False: If $f:D(0;1)\setminus\{0\} \rightarrow D(0;1)$ is analytic, then $0$ is not a pole ...
1
vote
1answer
21 views

Theorem regarding primitives and complex integration

In order to apply this theorem must $f$ be continuous in the entirety of the open set $\Omega$ or only on the curve $\gamma$?
0
votes
0answers
26 views

Complex singularity exponent

I am studying about complex singularity exponents of holomorphic functions. I need some help to clarify a few things: First, what a complex singularity exponent is, for the holomorphic function ...
1
vote
1answer
40 views

Computing the complex integral?

I am dealing with the following: $$\int_{0}^{\infty}\frac{x\sin(x)}{(z^2+a^2)(z^2+b^2)}dx$$ Furthermore, I know $a,b>0$ and I know $a\neq b$. I believe this is using Jordan's Lemma? I see that the ...
-3
votes
0answers
25 views

Integrating $\int_0^1\cos(\lambda x^3)dx$ using the saddle point method [on hold]

Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of saddle points along a certain contour. I am having trouble approaching this ...
0
votes
0answers
18 views

How to determine contours by looking at the exponential integrands?

I know that we determine the contours in contour integrals by looking at the exponential integrand (assuming there is indeed an exponential integrand in the given integral) but I don't know how. For ...
0
votes
0answers
30 views

How are singularities of complex functions classified?

I have the function $$\frac{z^2+1}{z^3+6z^2+z}$$ And I wish to find the residues at $z_0=0$ and $z_0=2\sqrt{2}-3$ because they are within my given contour. However, I am really confused when it comes ...
0
votes
2answers
28 views

Have I Correctly Defined the Set of Nonzero Complex Numbers $\mathbb{C^*}$?

If the set of complex numbers $\mathbb{C} = \{a+bi\mid a,b \in \mathbb{R}\}$, then what would be the definition of the set of nonzero complex numbers? Am I right in defining such a set as ...
0
votes
0answers
12 views

Is the Möbius inversion applicable in the case of number functions with values in $Q(x)$

I am looking for the cause of an erroneous calculation I did the details I cant present here. I guess a "Möbius inversion" I apply might be the cause. Normally the Möbius inversion is valid for ...
0
votes
2answers
58 views

Image of curves in the complex plane

I'm not really sure what I'm being asked in this question. If $x=C,y=C$ doesn't that mean $z=C+iC$?
1
vote
1answer
14 views

Showing a complex function is nowhere differentiable in a certain disc

I have a function and I am asked to prove that it is nowhere differentiable on an open disc. I found the cauchy riemann equations and saw that is is satisfied at the origin. I don't know what to do ...
0
votes
0answers
31 views

Counting zeros of an analytic function [on hold]

Suppose f is analytic on $Ball_R(0)$ and satisfies $|f (z)| < R$ for $| z| = R$. Using Complex analytic methods (such as Rouche's theorem), how can I find the number of solutions (counting ...
1
vote
1answer
34 views

Help With Bromwich Inversion Formula Proof

To prove(copied from handwritten notes so possibly wrong): Bromwich Inversion Formula. Fix $x_0∈ℝ $. If $F$ is complex analytic on $\{z:\Re z > x_0\} $ and for every $x>x_0$, $y↦ F(x + iy )$ ...
2
votes
2answers
41 views

Under what conditions do you use that $\operatorname{Res}{(f(z)/g(z))}=f(z_0)/g'(z_0)$?

In complex analysis, this seems to be a really helpful way to avoid having to expand out Laurent series. I am unclear, however, when it is appropriate to use this property. In specific, I'm worried I ...
0
votes
0answers
14 views

Laurent series expansion for powers of n?

I wish to expand the function: $$\dfrac{e^z}{z^n-c^n}$$ about the point $z_0=c$, where c is a constant greater than 0 and n is greater than 2. So I have that $e^{z-c}$ expands to ...
1
vote
1answer
29 views

Why is (-1)^(2/3) equal to -1/2+(i sqrt(3))/2

Can someone please explain to me how $(-1)^{\frac2 3}$ can be written as $\frac {-1}{2}+\frac{i \sqrt3} 2$ ? Do you use the corrolation $(-1)^c = e^{(i c \pi)}$, where ${c}$ is a constant?
3
votes
1answer
32 views

Zeros of polynomials with real exponents

Does every non-constant function of the form \begin{equation*} f(z)=a_0+a_1z^{r_1}+\ldots a_nz^{r_n} \end{equation*} have a complex zero? Here the $r_k$ are positive reals, the $a_k$ are arbitrary ...
2
votes
1answer
23 views

Double period except for poles

I'm trying to solve a problem in Complex Analysis whose function $f$ is defined in $\mathbb{C}$, is meromorphic and have double period $(f(z)=f(z+a)=f(z+b),\ \frac{a}{b} \notin \mathbb{R})$ except for ...
1
vote
2answers
25 views

Applying Cauchy Residue Theorem to $\int_{C}\frac{e^{z}}{sin^2{z} - 1}$

For $\int_{C}\frac{e^{z}}{sin^2{z} - 1}$, $C = \{|z|=3 \}$, this has singularities at $z = \frac{\pi}{2}$ and $z = \frac{3\pi}{2}$. So $Res(f,\frac{\pi}{2}) = \frac{e^{z}}{\sin(2z)} = ...
4
votes
1answer
30 views

Residues of $z^2\sin(\frac{1}{z})$

I must find the residues of $z^2\sin(\frac{1}{z})$ at $z = 0$. Since $z = 0$ seems to be an Essential Singularity, i'm not sure how I can continue to find the residue of the function. Usually I am ...
0
votes
1answer
23 views

Harmonic functions proof

I don't understand here why: $2(\Delta(u_x)^2+\Delta(u_y)^2) \geq 0$. Here $\Delta= \nabla^2, \quad u'_x=u_x $ etc
1
vote
4answers
31 views

Find Solution of trigonometric complex equation

Find the solutions of $\sin z = 3$ There are 2 ways to solve this, I know how to do this with: $\sin z = \frac{1}{2i}(e^{iz}-e^{-iz}) = 3$ Now, I am now doing in the way: $\sin z = \sin x \cosh y+i ...
8
votes
2answers
116 views

Dog Bone Contour Integral

Would someone please help me understand how to integrate $$ \ \int_0^1 (x^2-1)^{-1/2}dx\, ? $$ This is a homework problem from Marsden Basic Complex Analysis. The text book suggested using a "dog ...
0
votes
0answers
37 views

An entire function is zero

Suppose $f(z)$ is an entire function that have zero on positive integers. Does it follow $f$ is identically zero? This seems like an application of Liouville theorem. But I cant come with a function ...
2
votes
3answers
65 views

Is $(a+bi)(a-bi) = a^2 + b^2 $ solely a real number or a complex number?

I have not dealt with complex numbers for a while now, but I was wondering if I multiplied the complex number $a+bi$ by its conjugate $a-bi$ to obtain $$(a+bi)(a-bi) = a^2 + b^2 $$ where $a,b \in ...
2
votes
1answer
38 views

A problem about elliptic functions

I am trying to solve some problems in complex analysis, but I am not succeeding in the following problem. Suppose that $f$ is a function with the following properties: $f$ is non-constant; $f$ is ...
4
votes
1answer
60 views

A Funtional equation in Complex variables

I have been stuck on this problem for a long time : If $f(z)=u(x,y)+iv(x,y)$ , prove that a. $f(z)=2u(z/2,(-iz)/2) +$ constant b.$f(z)=2iv(z/2,(-iz)/2) +$ constant This result seems very ...
4
votes
2answers
35 views

$f=u+iv$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$?

$f(z)=u(x,y)+iv(x,y)$ holomorphic, $xu+yv = (x^2+y^2)e^x \cos y$, what is $f$? I tried to interprete $xu+yv$ as some part of a new function, for example, as the real part of $\overline{z}f$,but this ...
2
votes
0answers
38 views

$\frac{df}{dz}$ and $\frac{\partial f}{\partial z}$

If $f(z)=u(x,y)+iv(x,y)$, $z=x+iy$ what is the difference between $\frac{df}{dz}$ and $\frac{\partial f}{\partial z}$? I understand $\frac{\partial f}{\partial z}=\frac{1}{2}(\frac{\partial ...