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
11 views

Complex integration Cauchy Formula

$\oint_{\left | z \right |=0.5} \frac{dz}{(z-1)(\sin z)} $ Define: $f(z) = \frac{z}{(\sin z)(z-1)}$ Define: $g(z) = \frac{f(z)}{z}$ Now integrate using Cauchy Integration Formula $\oint_{\left | ...
-1
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0answers
9 views

Evaluate the given integral along the given (positively oriented) circle.

Ok, so I have the following problems that I am working on. It says to evaluate 1) where C is given by |z+1|=1/2 2) where C is given by |z-2|=1/2 3) where C is given by |z|=2 4) where C ...
0
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0answers
47 views

If $|f|+|g|$ is constant then each of $f, g$ is constant

Let $f,g: U \rightarrow \mathbb{C}$ be holomorphic on the open and connected subset $U$. If $|f| + |g|$ is constant on $U$ show that $f, g$ are constant on $U$. What can we say about finite or ...
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0answers
24 views

Is $z/\sin z$ analytic in the complex plane? [on hold]

Verify if the function $$f(z) = \frac{z}{\sin z}$$ is analytic in the complex plane?
2
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4answers
52 views

Complex Analysis book including integration

FOR BEGINNERS: Currently, I am looking for a textbook on complex analysis, which covers complex analysis from the beginning, and majorly focuses on contour integration, and the residue theorem. On ...
-2
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1answer
35 views

Evaluating a complex integral using the Cauchy integral formula [on hold]

I need to evaluate the following integral counterclockwise: $$\oint_{\left | z \right |=\frac{1}{2}} \frac{dz}{(z-1)\sin z} $$ using the Cauchy integral formula
0
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0answers
16 views

A function differentiable only at $0$ and for $|z|=1$

I need to find a polynomial function that is differentiable at the origin where $f'(0)=1$ and at every point $|z|=1$ but at no other point in the complex plane. I just have no clue how to solve ...
6
votes
1answer
251 views

Describing co-ordinate systems in 3D for which Laplace's equation is separable

Laplace's Equation in 3 dimensions is given by $$\nabla^2f=\frac{ \partial^2f}{\partial x^2}+\frac{ \partial^2f}{\partial z^2}+\frac{ \partial^2f}{\partial y^2}=0$$ and is a very important PDE in ...
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0answers
30 views

Properties of the Fourier transform

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. I want to show that $$ \widehat{g}(kn)= \widehat{h}(k), \\ \widehat{g}(l)=0, l \not\equiv 0 \ \text{mod} \ n.$$ I ...
0
votes
1answer
24 views

Having trouble combining Weierstrass approximation theorem and the infinite sequence of holomorphic functions

The Weierstrass approximation theorem says that all continuous functions on $[0,1]$ can be approximated uniformly by polynomials. Trying to facilitate the digestion of the fatty Christmas food, I ...
0
votes
4answers
52 views

If a continuous function satisfies $|f(z)^2-1|<1$ for every $z$, then either $|f(z)-1|<1$ of $|f(z)+1|<1$ for every $z$

Suppose a continuous function $f:D\rightarrow \mathbb{C}$ where $D$ is a plane domain, has the property $|f(z)^2-1|<1$ for every $z$ in $D$. Show that $|f(z)-1|<1$ of $|f(z)+1|<1$ for every ...
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0answers
17 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
0
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0answers
32 views

Complex analysis (Analytic function, sharp upper bound)

I encouter complex analysis problems the I think it is quite to do. Could anyone please give a hint or guideline. Thank you very much in advanced. Let $D$ be an open unit disc $\{z \in \mathbb{C}| ...
0
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3answers
94 views

Contour integral of $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$

I'm supposed to evaluate $\int_0^{2\pi} \frac{1}{A - cos \theta} d\theta$ Using a contour of a unit circle, $z=e^{i\theta}$. This is the same as: $$2i \oint \frac{1}{z^2 - Az + 1 } dz $$ The ...
5
votes
0answers
107 views

How can $ i $ be distinguished from $ - i $? [duplicate]

Mathematicians designate one solution to $x^2 = -1$ as $i$ and the other as $-i$. Would anybody notice if we switched their identities? Any polynomial $p(x)$ with a complex root will also have its ...
0
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2answers
62 views

Help Finding the Cauchy Principle Value of $\int_{0}^{2\pi}\frac{d\theta}{1+2cos(\theta)}$

$$\int_{0}^{2\pi}\frac{d\theta}{1+2cos(\theta)}$$ My attempt: parametrise using $z=e^{i\theta}$ (i think we always use a unit circle for CPV's) $\therefore dz = ie^{i\theta}d\theta$ $\implies ...
1
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1answer
71 views

$\sum |f_n|$ converges uniformly implies $\sum f_n$ converges normally

Is the following proposition true? Let $f_n \colon U\subset\mathbb{C}\longrightarrow \mathbb{C}$ be a sequence of continuous (or holomorphic) functions such that: $\sum |f_n|$ converges ...
4
votes
1answer
185 views

Conformal map of doubly connected domain into annulus.

I have a homework question that I'm stuck on. It asks Let $\Omega$ be a bounded domain whose boundary consists of two disjoint continua $C_1$ and $C_2$. Let $u(z)$ be the harmonic function on ...
1
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1answer
32 views

Images of lines $y = k = \mbox{constant}$ under the mapping $w = \cos (z)$

I want to solve this question: find the images of lines $y = k = \mbox{constant}$ under the mapping $w =\cos(z).$ I know that $w=\cos(z)=\cos(x)\cosh(y)-i\sin(x)\sinh(y)$ so $u=\cos(x)\cosh(y)$ and ...
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0answers
16 views

pole on the contour using the residu theorem, what is this formula of Plemelj?

I've tried solving the following problem but I get stuck at the very end... $f(z)$ is defined as $$f(z)=\frac{1}{(z-\alpha)^2(z-1)}$$ with $\alpha \in \mathbb{C}$ and $\operatorname{Im}(\alpha) ...
2
votes
1answer
66 views

When should I resort to Eulers identity?

I'm working on the following excercise: Calculate: $$\int_0^{+\infty} \frac{x^{\frac{1}{3}}\sin (x+\frac{\pi}{3})}{x^2+1}\operatorname dx$$ Using the contour-integral $\int_{\Gamma} ...
3
votes
2answers
46 views

Simply connected and connected in complex analysis

Can some one please help me with this, why is third set in the picture not simply connected. The definition of simply connected (in space of complex numbers) is: A set is said to be simply ...
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1answer
28 views

Checking where the complex derivative of a function exists

I have the following function: $$f(x+iy) = x^2+iy^2$$ My textbook says the function is only differentiable along the line $x = y$, can anyone please explain to me why this is so? What rules do we ...
0
votes
1answer
33 views

List all the elements of the subgroup of Möbius transformations preserving the set $\{0, 1 + i, \infty\}$

List all the elements of the the subgroup $M_{\{0, 1, \infty\}}$ of the group of Möbius transformations, preserving the set $\{0, 1, \infty\}$ and give an explicit isomorphism $M_{\{0, 1, \infty\}} = ...
2
votes
1answer
70 views

Complex Limit Without L'hopital's

I'm trying to solve for the limit of the following complex function as $z\to0$. I know L'hopital's rule but I'm to find the answer without using that method. The limit is: $$\lim_{z \to 0} ...
2
votes
1answer
34 views

Convergence implies Abel summability, and we only need to consider when $s=0$?

Suppose $\displaystyle c_n\in\mathbb{C}\textrm{ and}\sum_{n=1}^{\infty}c_n=s$. Then, prove $\displaystyle\lim_{r\to 1^{-}}\sum_{n=1}^{\infty}r^{n}c_n=s$. In my text, the author hinted that: we only ...
7
votes
2answers
2k views

Why is an integral of a complex function defined as a line integral?

In real analysis, we can define a line integral, but we also define (earlier) the regular definite integral. Why is it that in complex analysis we are interested only in a line integral?
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0answers
39 views

Proving Euler's formula without calculus

With $\cos x$ and $i \sin x$ in a complex plane, is there a proof that their sum is equal to $e^{ix}$ without resorting to calculus? All proofs I have found either directly proves the relation ...
3
votes
1answer
30 views

Is there any significance to complex function “monotone in norm?”

So, I was reading a question earlier where someone asked if something would be strictly monotone in the complex plane, and the comment was that this would be meaningless, since the complex numbers ...
14
votes
4answers
216 views

Evaluate $\int_1^\infty \frac {dx}{x^3+1}$

I would like some help with the following integral. I would like to find a contour line to evaluate $$\int_1^\infty \frac {dx}{x^3+1}$$ So one can see that on any circumference it goes to $0$, but ...
11
votes
2answers
135 views

Is there a good way to solve for z the equation $e^{i\pi} = e^{z\ln2} + e^{z\ln3}$?

$e^{i\pi} = e^{z\ln2} + e^{z\ln3}$ How can I deal with this? I want to solve for z. Does this help? $e^{z\ln2} + e^{z\ln3} = e^{z\ln2}(1 + e^{z(ln3-ln2)})$ If I write out z=x+iy then the ...
277
votes
7answers
7k views

“The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
-3
votes
1answer
40 views

What is the constant term of the Laurent Series for $\cos(z)/z^2$? [on hold]

What is the constant term of the Laurent Series for $\cos(z)/z^2$? I want to prove that the constant term from this series is $-1/2$.
1
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1answer
22 views

Finding the residue of a function with a surd variable

$$f(z) = \frac{1}{z-2\sqrt{z}+2}$$ Is this the correct way of doing this, please advice - Thanks, what i did was try to rationalise the expression first as follows: $$f(z) = \frac{1}{z-2\sqrt{z}+2} ...
1
vote
1answer
16 views

Elliptic functions $f(z+\lambda_1)=af(z) \; , \; f(z+\lambda_2)=bf(z) $

Let $\lambda_1$ and $\lambda_2$ be complex numbers with nonreal ratio. Let $f(z)$ be an entire function and assume there are constants $a$ and $b$ such that $$f(z+\lambda_1)=af(z) \;\;\;\;,\;\;\;\; ...
5
votes
2answers
56 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
0
votes
2answers
56 views

Why is it so difficult to use geometric methods to construct weight one modular forms?

I am trying to understand why it is that geometric methods of constructing modular forms of weight one for $SL(2,\mathbb Z)$ fail. I have a relatively rudimentary understanding of it, but it would be ...
2
votes
1answer
69 views

Show that $4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$

I am supposed to prove the following: $$4\frac{\partial}{\partial z}\frac{\partial}{\partial\bar{z}}=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\,\,\,$$ Using the definitons ...
7
votes
2answers
125 views

How can I prove $\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$

Can the residue theorem prove this? $$\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$$
1
vote
2answers
29 views

How is $ \lim_{z \to z_o} (z-z_o)\frac{f(z)}{g(z)} = \lim_{z \to z_o} \frac{f(z)}{g(z)-g(z_o)/(z-z_o)}= \frac{f(z_o)}{g'(z_o)}$?

I was reading this proof in Gamelin Complex Analysis (page 196): If $ f(z) $ and $ g(z) $ are analytic at $ z_o $ and if $ g(z) $ has a simple zero at $ z_o $ $$ Res[ \frac{f(z)}{g(z)},z_o ] = ...
10
votes
6answers
696 views

Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
1
vote
2answers
34 views

Comparison of the consequences of uniform convergence between the real and complex variable cases,

In the real variable case, I think that uniform convergence preserves continuity and integrability, i.e., for an integral of a sequence of continuous (or integrable) functions, which converge ...
3
votes
2answers
177 views

Prove that an analytic function on complex plane to its proper open and simply connected set is constant

Suppose that $V$ is an open, simply connected, proper subset of $\mathbb{C}$. Suppose that $f\colon\mathbb{C}\rightarrow V$ is holomorphic. Prove that $f$ is constant function. Give counter example ...
7
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0answers
75 views

Function's analytic continuation is its own derivative

This is the question we were asked at the university by our professor for complex analysis. Not as an exam, but as a challenge. I don't think he knew the answer himself. Find a nontrivial example of ...
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0answers
27 views

inequalities related to linear factional transformation and schwarz`s lemma

In both questions, it is said that I should use schwarzs lemma and linear factional transformation. But I don`t have any ideas how to use it. please give me some more hint
0
votes
0answers
42 views

What does complex square root as defined on Wikipedia look like: two questions

If you look at the third picture here, the surface representing the complex square root intersects the negative real axis at $0$. Later in the article the definition of the complex square root is ...
3
votes
1answer
79 views

how to compute this complex integral with high order polynomial?

Compute $$ \lim_{R \to \infty} \frac{1}{2\pi i} \int_{|z|=R} \frac{(2z^2+z-1)P'(z)}{P(z)+3}dz $$ where $P(z)=z^{10}+2z^9+z^5+1$. It seems like I may use residue but it contains too high ...
0
votes
1answer
35 views

Chain rule for the Wirtinger derivative $\frac{\partial}{\partial \overline{z}}$

I came over a calculation which used the identity $$ \frac{\partial(u \circ f)}{\partial \overline{z}} = \left( \frac{\partial u}{\partial \overline{z}} \circ f \right) \overline{f'}. $$ I tried to ...
0
votes
1answer
19 views

Bounded functions composed with Möbius maps

Hopefully easy question here: What is the most succinct method/technique to prove the following statement?: Let $u \in L^{\infty}(\Bbb D)$. Show that $||u(\varphi_{z})||_{\infty}$ is ...
0
votes
0answers
28 views

How to prove that $f$ can be expressed as a ratio of polynomials, given that $|f(z)|=1$ when $|z|=1$? [duplicate]

Given: $f$ is analytic in $| z|\leq1$ and $|f(z)|=1$ when $|z|=1$. Prove that $f(z)=P(z)/Q(z)$ where $P$, $Q$ are polynomials.