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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help contour integral and residue theorem

compute $$J= \int_\infty^\infty \frac{dx}{x^3-8i}$$ simplify your answer until you get a purely imaginary number When I initially started this question, I put $z^3$ and $8i$ into polar form and ...
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1answer
39 views

Differential form on complex torus

Suppose $T= \mathbb{C}^n/\Gamma$ is a $n$-dim complex torus. How to prove that every exact $2$-form which has no $(0,2)$ component must be the image of a $(1,0)$ form? Is every torus Kahler? If the ...
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10 views

Computing ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$ for $U(z)= \int_\gamma (z - a)^n\ dz$

Goal: Let $$ U(z)= \int_\gamma (z - a)^n\ dz $$ I'm trying to compute ${\partial U \over \partial x}$ and ${\partial U \over \partial y}$. Attempt: I know that $(z-a)^n$ is the derivative of ...
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1answer
20 views

How to know that $(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$

How to know that $$(z-z_0^3)(z-z_0^5)(z-z_0^7) = \sum_{k=0}^3 z^k z^{3-k}_0$$ with $z_0$ a root of $z^4+1$. I can check that it is true, but is there a way to tell, by seeing the LHS expression, ...
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38 views

Deducing Laplace Formulas

I have to compute the followings integrals $\forall\; b\in \mathbb{C},\; \text{Re} \;b \gt0,p\gt 0$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x-ib}$$ $$ \int_{-\infty}^\infty \frac{e^{ipx}}{x+ib}$$ ...
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37 views

Let $f(z):=e^{\frac 1 z}, {z \in \mathbb C \setminus\{0\}}$.What values of $z$ are $ f(z)=re^{i\phi}$ for $r\in(0, \infty), \phi\in\mathbb R$?

Consider $f(z):=e^{\frac 1 z}, {z \in \mathbb C \setminus \{0\}}$. For which values are $f(z)$ real ? I've considered $e^{\frac 1 {a+ib}} = e^{\frac {a-ib} {a^2+b^2}}$. For which values are $f(z)$ ...
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0answers
15 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
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49 views

Any general methods to calculate integral of $P(x)/Q(x)$ from $0$ to $\infty$?

In complex analysis, we have general formula for $P(x)/Q(x)$ [$P$ and $Q$ are polynomials] from minus infinity to infinity, if $ \deg Q - \deg P > 2$. Is it possible to have a general formula for ...
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18 views

integrating of complex exponential function

I know $\int x^a dx=\frac{x^{a+1}}{a+1}$ when $a$ is real. How I can calculate this integral when $a$ is complex?
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48 views

Changing research area in grad school

I'm a PhD student about to close out my third year. My current research area is operator algebras. At the beginning of this semester I completed my qualifying exams (this was accomplished a semester ...
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1answer
41 views

Radius of Convergence of a Complex Taylor Series

I've recently been doing some complex analysis questions and come across a few of this type: Find the radius of convergence of the Taylor series at $z=-1$ of the function ...
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0answers
13 views

linear fraction trasformation automorphism

i have a question which is also one of my homework problem I could not solve. let $a,b,c,d$ are real numbers and $ad-bc>0$ then $f(z)=az+b/cz+d$ is a $Aut(H)$, $H$ is the upper-half plane. I ...
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2answers
52 views

Complex Analysis: The Identity Principle

I'm studying some complex analysis at the moment and have come across the Identity Principle. The statement is as follows: If the function $f$ is holomorphic in a connected subset ...
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1answer
22 views

Convergence Question:

This is related to the Dirichlet eta function. Does $$\int_1^\infty \frac{dx}{x^z}$$ converge for $Re(z)>1$? Just wondering. If so, then does $$\int_1^2 \frac{dx}{x^z}+\int_3^4 ...
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10 views

normality of set of analytic functions whose derivative is normal

I have this question from old preliminary exam problem set. (a) Show that if F⊂H(G) is normal then F′={f′:f∈F} is also normal. (b) Does F⊂H(G) normal imply F′={f: f′∈F} is normal? Otherwise give a ...
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1answer
27 views

Relation between continuity of $f$ and analyticity of $f(z)^8$

If $f(z)$ is continuous on some domain $D$ and $f(z)^8$ (the function to the eighth power, not the eighth derivative) is analytic, then why does this imply that f is analytic on a neighborhood of each ...
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1answer
70 views

Is f(z)=1/z truly an analytic function

For an analytic function $f(z)$, we have $$\frac{\partial f}{\partial \bar{z}}=0.$$ Consider the function $f(z)=\frac{1}{z}$, which, at first sight, is a bona fide analytic function. However, we can ...
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48 views

Does maximal principle imply open mapping theorem for any continuous function?

At first I spent a lot of time looking for counterexamples because I had never seen such a claim that M.P. implies O.M.T.. But later I realized the claim might be true, so I just had a try and proved ...
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3answers
49 views

Algebraic Equation?

$$Ve^{i\theta} = We^{i\phi}$$ where, $V$ and $W$ are some real constants. From this my book concludes: $\theta = \phi$. How does it conclude this? I don't see why its valid to just equate the ...
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56 views

Help with contour integral!

Question: Consider the integral $$I_N = \frac{1}{2\mathrm{i}\pi} \oint_C \sin\frac{2z}{N}\cot (z)\,\mathrm{d}z $$ where $C$ is the rectangle with corners $\frac{\pi}{2} \pm \mathrm{i}$ and ...
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1answer
23 views

Verifying a condition for which $\int_\gamma p\ dx + q\ dy$ depends only on endpoints

Hypothesis: Suppose there exists a function $U(x,y)$ in $\Omega$ with partial derivatives $${\partial U \over \partial x} = p \quad \quad {\partial U \over \partial y} = q$$ Goal: Show that the ...
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+250

Formula for decomposing a form into $(p,q)$ forms

Let $L: \mathbb{C}^n \to \mathbb{C}$ be a real linear map. In other words, $L(a\vec{v}_1+b\vec{v_2}) = aL(\vec{v}_1)+bL(\vec{v}_2)$ for all $a,b \in \mathbb{R}$. Then $L$ decomposes uniquely into a ...
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21 views

Image of $\{z:|\mathfrak{Re}(z)|<1\}$ under $f$

If I have a set of complex numbers, $S=\{z:|\mathfrak{Re}(z)|<1\}$ and I apply the mapping $$f:\mathbb{C}\rightarrow\mathbb{C}$$ $$f(z)=(1+i)z+1$$ How can I write the image set? I know that the ...
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52 views

How to prove $\frac{\partial z}{\partial \bar{z}} = 0$ if and only if $\frac{\partial \bar{z}}{\partial z} = 0$

Here is a problem form my complex analysis HW. Unfortunately, I really have no idea how to go about this. Specifically, I don't really know how to take partials of that form. Does anyone have ...
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0answers
16 views

Clarification on the absolute convergence of Mellin transform

I have a question I haven't been able to find a direct answer to that I presume is true but I am unable to show. We know these two following results on the mellin transform. If $$\int_0^\infty ...
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2answers
57 views

The derivative of $\rho e^{it}$

Why is $${df \over dz} \rho e^{it} = i \rho e^{it} \text{?}$$ The product rule states that $$ {df\over dz}(f_1 \cdot f_2) = f_1 f'_2 + f'_1 \cdot f_2 $$ so why doesn't this imply that $$ ...
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31 views

Can the hypergeometric function be extended analytically to the complex plane in the interval [1,$\infty$ )?

Just a thought. The hypergeometric function, which can be written as: $$F(a,b,c \space;z) = \frac{\Gamma (c)}{\Gamma (b) \Gamma (c-b)}\int_0^1t^{b-1}(1-t)^{c-b-1}(1-zt)^{-a}dt$$ is obviously ...
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1answer
42 views

Question on the Prime Number Theorem (the Tchebychev Function) [duplicate]

This has been giving me nothing but a headache: Let the Tchebychev Function, $\psi (x)$ be defined: $$\psi (x) = \sum_{p^m \le x}\log p \space \space \space , \space \space \space p \in \mathbb P$$ ...
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0answers
23 views

What is meant by a “general line integral of form $\int_\gamma p\ dx + q\ dy$”?

In his text on complex analysis, Ahlfors speaks of "general line integrals of form $\int_\gamma p\ dx + q\ dy$". I'm curious exactly what is meant by this. I take it that $p$ and $q$ are not ...
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49 views

What is the difference between a removable singularity and an essential singularity

I am learning about singularities, zeros and poles in complex analysis. I am still a bit rough of the definitions and I am not sure of the difference between a removable singularity and an essential ...
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32 views

Showing $f(z_0) = \frac{1}{2 \pi} \int_0^{2 \pi} f(z_0+Re^{i\theta}) \ d\theta$

Suppose that $f$ is analytic on and inside the circle $|z-z_0|=R$. Show carefully that $f(z_0)$ is equal to the average of $f$ on ${|z-z_0|=R}$, i.e. show that $$f(z_0)={1 \over {2 \pi}} \int_0^{2 ...
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1answer
46 views

How prove this $f(z)=0$ if $f(z)$ is entire function

let $f(z)$ be an entire function such that $$\int_{0}^{2\pi}|f(re^{i\theta})|d\theta\le r^{16/5}, \quad \forall r>0$$ show that $$f(z)=0$$ Thank you for you help.maybe this can use Cauchy theorem ...
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20 views

Conformal equivalence of resistance

I'm currently working on a system that uses a logarithmic and a Schwarz-Christoffel transformation to calculate the resistance of a specific area. With resistance I mean the resistance that would ...
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1answer
23 views

How to prove that $\pi \frac{e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}}}{1 - e^{2\pi it}} = \frac{\pi}{2\cos\left(\frac{\pi t}{2}\right)}$

How to prove that : $$\pi \frac{e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}}}{1 - e^{2\pi it}} = \frac{\pi}{2\cos\left(\frac{\pi t}{2}\right)}$$ I start with $$e^{it\frac{\pi}{2}}-e^{it\frac{3\pi}{2}} = ...
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0answers
17 views

Computing a complex line integral $dz$ in terms of line integrals $dx$ and $dy$

Goal: I'm trying to verify the calculation claimed by Ahlfors that $$\int_\gamma f(z)\ dz = \int_\gamma (u\ dx - v\ dy) + i \int_\gamma (u\ dy + v\ dx)$$ Attempt: $$\int_\gamma (u\ dx - v\ dy) + i ...
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1answer
22 views

Find a conformal map from the exterior of the closed unit disk to the unit disk

Question: Find a conformal map from the exterior of the closed unit disk to the unit disk. Also, prove that it is indeed a conformal map (bijective and holomorphic along with its inverse). I missed ...
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9 views

potential from conformal mapping

I have this following question unable to solve: Use the transformation $w=z+\frac1z$ to find the electrostatic potential at any point when a conducting cylinder of unit radius is placed in a ...
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23 views

Analytic functions theorem

There is a result on page 175 in the book "All the Mathematics You Missed" by Thomas Garrity. A complex-valued function $f(x,y)=u(x,y)+iv(x,y)$ is analytic at the point $z_{0}=x_{0}+iy_{0}$ if and ...
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74 views

Why is $sinx$ the imaginary part of $e^{ix}$? [duplicate]

Most of us who are studying mathematics are familiar with the famous $e^{ix}=cos(x)+isin(x)$. Why is it that we have $e^{ix}=cos(x)+isin(x)$ and not $e^{ix}=sin(x)+icos(x)$? I haven't studied Complex ...
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1answer
34 views

Counterexample for Hartogs' Extension Theorem

I'll refer to Hartogs' Extension Theorem as it is stated in Wikipedia (https://en.wikipedia.org/wiki/Hartogs%27_extension_theorem#Formal_statement). I am trying to find a counterexample to show that ...
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30 views

Limit with complex numbers

We're dealing with complex analysis problem: $$\lim_{z\rightarrow 2i}\frac{z^5-4iz^4-4z^3+z^2-4iz+4}{5z^4-20iz^3-21z^2-4iz+4}$$ Would it be $\frac{1}{5}$?
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10 views

Logarithms and arguments - counting zeros and poles with logarithmic derivatives

Intuitively trying to understand the argument principle, why does the integral of the logarithmic derivative of a meromorphic function over a closed curve give the number of zeros minus the number of ...
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2answers
47 views

Can you Simplify This Complex Expression?

Let $a,b\in\mathbb{R}$ and $i=\sqrt{-1}$. Does the expression $(a+i b)^{1/3} + (a-i b)^{1/3}$ simplify to a real valued expression defined solely in terms $a$ and $b$?
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1answer
41 views

Find the integral in the complex plane

I'm having some trouble computing these integrals, they're on the practice final, but no solutions given. I'm hoping to get some help here. Calculate the following Integral of $(z \cdot ...
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23 views

Runge Kutta stability region for forward euler and explicit midpoint

The interval of absolute stability is the intersection of the region of absolute stability in the complex plane with the real axis.Show that Runge Kutta forward Euler and RK explicit midpoint have the ...
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3answers
51 views

Radius of convergence of ${\sum_{n=0}^{\infty}} \frac{z^{2n}}{4^n}$

How to calculate the Radius of convergence of $\displaystyle{\sum_{n=0}^{\infty}} \frac{z^{2n}}{4^n}$ Can we use the Root test? How?
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1answer
50 views

Solve the Dirichlet problem [on hold]

Solve the Dirichlet problem on D , with boundary-value data h(θ) = cos(θ), θ ∈ [0, 2π] I'm supposed to solve using the Poisson Kernel but I am completely lost
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1answer
24 views

Pole of function $f(z) = \frac{1}{z^4 + a^4}$ [on hold]

Let $$f(z) = \frac{1}{z^4 + a^4}$$ Show if $z_v$ is pole of $f$ then $Res(f,z_v) = - \frac{z_v}{4a^4}$.
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0answers
26 views

Decide if a complex function is holomorphic

Problem Analyze where the function of $z=x+iy$ is holomorphic and find $f'(z)$ $f(z)=x^2-y^2-2xy+i(x^2-y^2+2xy)$ I know that the function must satisfy the Cauchy-Riemann equations in order to be ...
1
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2answers
64 views

Complex numbers confused!!

If you give me a complex number say $z=2+3i$, then I can easily find $\text{Im}(z)=3$ and $\text{Re}(z)=2$ but when this polar coordinates stuff came, I lost my head! So say ...