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
25 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 ...
1
vote
1answer
251 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
48 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
26 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+(z-c)+\frac{1}{...
1
vote
1answer
43 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
86 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
36 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
35 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)} = \frac{e^{\...
4
votes
1answer
39 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
28 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
65 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 ...
10
votes
2answers
450 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 ...
2
votes
4answers
122 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
84 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
894 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 ...
5
votes
2answers
82 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
47 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 f}{\...
2
votes
1answer
186 views

Holomorphy of continuous function on $(A\cup B)^C$

Let $f:\mathbb C\to \mathbb C$ be a continuous function and $A,B\subseteq \mathbb C$ two open connected sets with $\overline{A \cup B}=\mathbb C$. Further, we know that $f\mid_A$ and $f\mid_B$ are ...
3
votes
2answers
105 views

calculate the principal part of $\tan(z)$ at $\frac{\pi}{2}$

calculate the principal part of $\tan(z)$ at $\frac{\pi}{2}$. of course $\tan(z) = \frac{\sin(z)}{\cos(z)}.$ Because $\cos(z)$ is of order 1 in $\frac{\pi}{2}$ we know that our primal part must look ...
1
vote
1answer
41 views

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$

Let $C=\partial D_1(\mathbf i/2)$, compute $\int_C\frac{dz}{z^2+1}$ $C=\partial D_1(\mathbf i/2)$ is the boundary of the disc with center $\mathbf i/2$ and radius $1$, then $\mathbf i$ is contained ...
0
votes
3answers
54 views

Show that $\sum_{n \neq 0} \frac{(-1)^{n+1}}{in} e^{in\theta} = 2 \sum_{n=1}^\infty (-1)^{n+1} \frac{\sin n\theta}{n}$.

Show that $$\sum_{n \neq 0} \frac{(-1)^{n+1}}{in} e^{in\theta} = 2 \sum_{n=1}^\infty (-1)^{n+1} \frac{\sin n\theta}{n}.$$ This is not an exercise. It is an example from Stein, Fourier Analysis - ...
1
vote
0answers
37 views

Confusion between partial and straight derivative wrt z

If $f(z)=u(x,y)+iv(x,y)$, $z=x+iy$ under what conditions is $f_z=\frac{df}{dz}$?
1
vote
1answer
58 views

Evaluation of $\prod_{k=1}^{\infty}\frac{a+k^2}{b+k^2}$

While playing around with the question The convergence of a sequence with infinite products, I found Mathematica to give me the result $$ \prod\limits_{k=1}^{\infty}\frac{a+k^2}{b+k^2} = \frac{\sqrt{...
0
votes
0answers
22 views

What is a bounded sequence of holomorphic functions?

Let $\Omega\subseteq\Bbb C^n$ open, $\{f_n\}_n\subseteq\operatorname{hol}(\Omega,\Bbb C)$ bounded. What does this mean? A numerical sequence $(a_n)_n\subset\Bbb C$ is bounded if $\exists M>0$ s....
1
vote
1answer
121 views

Prove that there is no function $f$ that is analytic. [duplicate]

Prove that there is no function $f$ that is analytic in $\mathbb{C}\setminus\{0\}$ and satisfies $$|f(z)|\geq\frac{1}{\sqrt{|z|}},\quad \operatorname{for all}\quad z\in\mathbb{C}\setminus\{0\}$$ I am ...
1
vote
2answers
53 views

How to show real analyticity without extending to complex plane

Suppose we have some $f \in C^\infty(\mathbb{R},\mathbb{R}).$ For example, $$f(x)=(1+x^2)^{-1}.$$ Using complex analysis, we can easily show $f$ is real analytic. Is there an easy, general method ...
1
vote
0answers
30 views

Cauchy's Integral Formula: Where have I gone wrong?

I have the function $$M(\mathbf{r})=\frac{\pi}{2}\left(erf\left(9-\lvert \mathbf{r} \rvert\right)+1\right)$$ where $erf(x)$ is the usual error function. Since $\mathbf{r}$ is $\in \mathbb{R^2}$, I ...
2
votes
2answers
50 views

Show that $|e^z -1| \leq e^{|z|}-1$ for any z

Show that $|e^z -1| \leq e^{|z|}-1$ What i have tried is Let $z=x+iy$.Then, $$|e^z-1|=|e^x\cos y-1+ie^x\sin y|=\sqrt{(e^x\cos y-1)^2+(e^x \sin y)^2}=\sqrt{e^{2x}-2e^x\cos y+1}$$ I stuck here and ...
0
votes
1answer
66 views

Parameterizing $C$ in the complex plane.

Let $C$ be the boundary with vertices at the points $0,3i,-4.$ Is the following parameterization correct? $C_1:z_1(t) = it, 0 \leqslant t \leqslant 3,$ $C_2=z_2(t) = 3i(4-t)-4(t-3), 3 \leqslant ...
1
vote
2answers
180 views

Laurent Series, Taylor Series, and Order of Poles. A tale of confusion.

For $\int_{C}\frac{\sin(z)}{(z^2 + 2z - 3)^2} dz$, where $C = \{|z|=2\}$, we have singularities are $z = -3$, $z = 1$. So only $z = 1$ is contained within the contour. This singularity has order $m=2$ ...
0
votes
2answers
27 views

Finding the value of $I=\int_C \overline{z} dz$ along $|z|=2$ from $z=-2i$ to $z=2i$

I want to find the value of the integral $$I=\int_C \overline{z} dz$$ When $C$ is the right hand half of the circle $|z|=2$ from $z=-2i$ to $z=2i$ Refer to beautifully made picture: Now I am new to ...
0
votes
1answer
41 views

Improper integral (using methods in complex variables) [closed]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
6
votes
1answer
140 views

Method of Steepest descents integral

I am looking to evaluate the following asymptotic integral: Find the leading term of asymptotics as $\lambda\to\infty$ $I(\lambda)=\int_0^1\cos(\lambda x^3)dx$ Using method of steepest descents ...
1
vote
1answer
106 views

Expand a function into a Laurent series about a point?

Take the function $f(z)=(z^2+3z+2)e^\frac{1}{z+1}$ We want to expand this into its Laurent series about $z_0$=-1. Alright, so I'm a little confused. This converges everywhere but -1, which throws me ...
1
vote
0answers
35 views

Beyond the Basics:Complex Analysis Topics/textbooks Suggestions

I am currently taking a semester long Graduate course in Complex Analysis. We have covered Basics of Complex Analysis,Automorphisms of Disc and Upper Half Plane,Riemann Mapping Theorem,Weierstrass and ...
2
votes
1answer
75 views

Complex integration on circle

Calculate the integral of $g(z)$ along the closed path $|z-i|=2$ in the positive direction when i)$g(z)=\frac{1}{z^2+4}$ ii)$g(z)=\frac{1}{(z^2+4)^2}$ First I checked the described area $$|z-i|=2\...
0
votes
1answer
105 views

Stationary Phase approximation of $\dfrac{1}{\pi}\int_0^{\pi}\cos(x\sin\theta-n\theta)d\theta$ (Bessel Function)

I'm trying to approximate $$\dfrac{1}{\pi}\int_0^{\pi}\cos(x\sin\theta-n\theta)d\theta$$ Where x goes to infinity I know to make it complex and then use the small angle approximation for $\sin\theta$...
0
votes
1answer
29 views

Complex integration and theorems

If $C$ is a closed path oriented in the positive direction and $$g(z_0)=\int_C \frac{z^3+2z}{(z-z_0)^3}$$ show that $g(z_0)=6\pi iz_0$ when $z_0$ is in interior of $C$ and $g(z_0)=0$ when $z_0$ is out ...
3
votes
2answers
64 views

If $\lim\limits_{z \to \infty} p(z) = \infty$, then $p(z)$ is a constant

Claim: If $p$ is an entire function and $\lim\limits_{z \to \infty} p(z) = \infty$ and $p(z) \neq 0$ $\forall z \in \Bbb C$, then $p(z)$ is a constant. Proof: Define $f(z) = \frac{1}{p(z)}$ so $...
0
votes
1answer
116 views

Complex function, analyticity domain

Find the function domain of analyticity i)$f(z)=\frac{z^2}{z-3}$ ii)$f(z)=ze^{-z}$ To check the domain of analyticity of a function, I only need to replace $z=x+iy$ and check the conditions of ...
2
votes
1answer
91 views

Existence of holomorphic function.

How to determine whether for given $a,b,c,d(reals)$ there exists a holomorphic $f:D\to D$ with $f(a)=b$ and $f ′(c)=d$ , where $D=\{|z|<1\}$. For example does there exist a holomorphic function $...
1
vote
1answer
88 views

Holomorphic equivalent to analytic

A holomorphic function is differentiable everywhere and satisfies Cauchy-Riemann condition. Prove that a function is holomorphic if and only if it's analytic? I have no idea how to prove this. ...
2
votes
1answer
44 views

Determine all analytic $f$, wherefor $|f(z)| \leq C(|z|^ {3/2} + |z-1|^{-3/2})$ on $\mathbb{C}\backslash\{1\}$ for some $C>0$.

Determine all analytic $f$, wherefor $|f(z)| \leq C(|z|^{3/2} + |z-1|^{-3/2})$ on $\mathbb{C}\backslash\{1\}$ for some $C>0$. In the assignment f needs to have the following property as well: $f(...
1
vote
2answers
82 views

Find an analytic function $f:\mathbb{C}\setminus\{-1\}\rightarrow \mathbb{C}$ such that $f'(z)=\frac{z}{z+1}$ or show that no such function exists.

I have a guess that the function does not exist. But I dont know how to show it. I have been suggested to look at the following theorems: 1): If f is entire, then f is everywhere the derivative of an ...
2
votes
1answer
101 views

Is there a measure invariant with respect to the Möbius transformation?

I would like to use a measure ${\rm d} \mu (z)$ on ${\mathbb C}$ so that for any $f(z)$ $$\int_{\mathbb C} f(z) {\rm d} \mu (z)$$ is invariant under Möbius transformations. Taking the ...
2
votes
1answer
120 views

Some issues with proving that a sequence is convergent

I recently tried (in the sense that I believe the thesis holds) to prove that, given $a\in\mathbb{R}^+$, there exists $$\lim_{n\rightarrow+\infty}\sqrt[n]{\sum_{k=0}^{\lfloor n/5\rfloor}{n-4k\choose ...
0
votes
2answers
28 views

absolute value of the addition of two complex numbers

I am working on a problem, and along the way have to take $|a-z|^2$, where $a$ and $z$ are complex. I know the triangle inequality, but I am trying to find a formula for $|z_1-z_2|$ = ?
0
votes
1answer
39 views

Finding a Z for which we can show that i/-i is a branch point

I have been given the following formula: $$ f(z) = \sqrt{(z^2+1)} = \sqrt{(z+i)(z-i)} $$ And I have to prove that i and -i are two branch points: if you make a circle around either of those points in ...
0
votes
0answers
95 views

Proof that principal branch of logarithm is continuous function

I tried to do an exercise in my book, please could someone tell me if my thoughts are correct? Exercise: Prove that $\arg : \mathbb C_- \to \mathbb C$ is continuous and conclude that the principal ...
1
vote
1answer
44 views

Why is $\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$?

Why is $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| \leq \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|$$ and not $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$