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, ...

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1answer
33 views

Explain a complex number identity

A college math instructor provided me with the following: $$\left|1+e^{ix}\right|^2=\left(1+\cos x\right)^2+\sin^2 x$$ Can anyone show me how this is done?
1
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1answer
19 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 ...
0
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3answers
45 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
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0answers
27 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
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1answer
33 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} = ...
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0answers
18 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 $\exists M>0$ ...
1
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1answer
67 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
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2answers
44 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
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0answers
24 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
40 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
17 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 ...
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0answers
13 views

equvivalence resistance of hexagonal infinite

I am trying to evaluate equivalence resistance between two nodes of hexagonal infinite grid, I am stuck at the integral at end of the image attached. pl see if the integral could be evaluated. Let ...
1
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2answers
30 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
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2answers
13 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
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1answer
27 views

Improper integral (using methods in complex variables) [on hold]

Let $0<a<1$. Evaluate the integral $$\int_0^\infty \frac{x^{a-1}}{1+x} dx.$$
3
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1answer
53 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 ...
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votes
2answers
22 views

Describe the image of the strip 0 < x < 1 under the mapping w=z/(z-1)? [on hold]

I heard this transformation would be mapped to either a circle or a line.
1
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1answer
43 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 ...
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0answers
24 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
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1answer
30 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 ...
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0answers
39 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 ...
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1answer
24 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
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0answers
58 views

Arguing that a complex function tends to zero fast enough to ensure that $\lim_{N \to \infty} \int_{C} f(s) \, ds = 0$

Consider the complex function $$ f(s) = \frac{\Gamma(a+s)}{\Gamma(b+s)}\frac{z^{s}}{\sin (\pi s)}$$ where $|z| <1$ and $- \pi < \arg(z) < \pi$. Let $C$ be the right half of the circle $|z|= ...
3
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2answers
56 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 ...
2
votes
0answers
32 views

Existence of analytic function. [on hold]

How to prove that there exist a holomorphic function $f:D \rightarrow D$ such that that$ f(3/4)=3/4$ & $f′(2/3)=3/4.$ where $\{D=|z|<1\}$. And there exist a holomorphic function $f:D ...
0
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0answers
19 views

when is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t) \, dt = 0$, with $x \in \mathbb{R}$?

For what value of $x \in \mathbb{R}$ is $\int_{t=-x}^{x} e^{iat}\operatorname{sinc}(t)\,dt = 0$, where $a$ is some constant?
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1answer
25 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
37 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 ...
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1answer
36 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
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1answer
36 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: ...
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votes
0answers
21 views

Integral of $\frac{1}{z^2+4}$ [on hold]

How do I calculate the integral of $\frac{1}{z^2+4}$ from $3i-2$ to $i+2$ clockwise round the circle $|6i-z+2|^2=25$? I think we're supposed to use Cauchy's integral formula, but it's not really been ...
1
vote
2answers
66 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
28 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 ...
1
vote
1answer
78 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 ...
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votes
2answers
20 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
0answers
22 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 ...
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0answers
18 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
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1answer
37 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 $$\left| \int_{|z|=1} \frac{f(z)}{z}dz\right| = \int_{|z|=1} \frac{|f(z)|}{|z|}|dz|?$$
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0answers
14 views

Complex analysis book recommendation for electrical. eng.

I need recommendation about complex analysis book. As I'm electrical eng. student, it should cover everything one engineer need to know about that mathematical field, but without strict mathematical ...
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0answers
18 views

Prove that $(n^2 )f(1/an)$ is bounded. [duplicate]

f is analytic from $\mathbb{C}\to \mathbb{C}$. Let $mod[f(\frac{1}{n})]\le\frac{1}{n^{3/2}}$ for each $ n\in\mathbb{N}$. Then prove that $(n^2 )f(1/an)$ is bounded. I just need a brief idea how ...
0
votes
1answer
38 views

Solve Trigonometric Complex Equation

Find all solutions of $\sin (z) = 2$. Here are the things I did: 1) By definition: $\sin z =\dfrac{e^{iz} − e^{−iz}}{2i}= 2$. Multiply $2i$ to the equation and make it quadratic: $e^{2iz} ...
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1answer
30 views

Order of a pole and the Residue

I have $\int_{C}\frac{e^{z} - 1}{z^6}$. The contour is irrelevant at the moment. I initially thought the order of the singularity of $z = 0$ would be $6$, but it turns out is $5$. Because $$e^z - 1 ...
2
votes
1answer
24 views

function constant on arc is constant on boundary

I was reading the answer to this question: on the boundary of analytic functions This answer makes sense to me up until the last line. What does having isolated zeros have to do with $f$ being ...
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0answers
16 views

Please could someone check my answer to this exercise about complex power?

I tried to solve the following exercise, please could someone check my answer and tell me if it is correct? Exercise: Calculate $\{a^b \} := \{ \exp(b \log |a| + i b \arg a) \exp (2 \pi i b k) \mid ...
0
votes
1answer
29 views

Could someone please check my result for this complex logarithm exercise?

I solved the following exercise and would appreciate it if someone would check my answer: Exercise: Find the principal value of the logarithm of the following numbers: $(i(i-1))^i$ and $i^i (i-1)^i$ ...
0
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0answers
28 views

Lebesgue integration in $\mathbb{C}$

I'm confused as to how we are supposed to integrate $$\frac{1}{\pi}\int_U\left[\frac{d}{dz}\left( \frac{z-\alpha}{1-\bar\alpha z }\right)\right]^2 \, dm$$ where $U$ is the unit disc, ...
0
votes
0answers
33 views

Why does the Radius of Convergence prove the fundamental theorem of algebra?

The radius of convergence of $ \sum a_n (x-x_0)^n $ is given by $$ \frac{1}{R} = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n}\right| $$ if the limit exists in the extended reals. [Proof of ...
1
vote
1answer
18 views

Clarification when using Mean Value Property to prove Fundamental Theorem of Algebra

We say that $f$ satisfies the Mean Value Property (MVP) on a ball $B(a,R) = \{z; |z-a| <R \}$ if $$ f(a) = \frac{1}{2 \pi} {\int_0}^{2\pi} f(a + te^{i \phi}) d \phi$$ for $0 < t <R.$ It is ...
1
vote
1answer
22 views

Cauchy integral formula for rational function, help with step

I have $P(\lambda) = (i\lambda)^m + O(\lambda^{m-1})$ a polynomial in $\lambda$, and $\Gamma$ a contour counterclockwise around the roots of $P$. I need to prove: ...
0
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0answers
14 views

Sequence of the derivative of converging trigonometric polynomials…

Suppose $\Gamma$ is a closed curve in the plane , with parameter interval $[0,2\pi]$ . Take $\alpha \notin \Gamma$ . Approximate $\Gamma$ uniformly by trigonometric polynomials $\Gamma_n$ . Show ...