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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13 views

Proving existance of a linear functional

Let (X, ||.||) be a normed space, and let A, B ⊂ X be disjoint convex sets such that B is closed and A is compact. Prove that there exists $ϕ ∈ X^*$ such that $sup_{a\in A} Reϕ(a) < inf_{b\in B} ...
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0answers
11 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 ...
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0answers
8 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 ...
4
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0answers
90 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 ...
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1answer
28 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
15 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) ...
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1answer
63 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} ...
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1answer
27 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 ...
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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 ...
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2answers
57 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 ...
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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 ...
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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 $-1/2$.
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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 ...
0
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1answer
31 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\}} = ...
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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} ...
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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) \;\;\;\;,\;\;\;\; ...
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2answers
55 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 ...
3
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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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2answers
28 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 ] = ...
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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 ...
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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 ...
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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
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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.
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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 ...
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0answers
17 views

how to prove that holomorphic function mapping complex onto complex is linear? [duplicate]

How to prove that any one to one holomorphic function mapping complex plane onto itself is linear?
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1answer
25 views

Help with simplification of an expression

I was solving the residues of $f(z)e^{zt} = e^{zt}\frac{\ln(z)}{z^2+1}$ as follows: $$\operatorname{Res}(f(z)e^{zt}, i) = \lim_{z\to i} (z-i)\frac{e^{zt}\ln(z)}{(z-i)(z+i)} = ...
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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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2answers
43 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
7
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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? $$\sum_{n=1}^{\infty }\frac{1}{n^3(n+1)^3}=10-\pi ^2$$
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36 views

Need help with holomorphic functions on a domain interval removed.

I want to prove that for a region $\Omega$ with interval $I=[a,b]\subset\Omega$, if $f$ is continuous in $\Omega$ and $f\in H(\Omega-I)$, then actually $f\in H(\Omega)$. Is this problem related to ...
6
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1answer
103 views

Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle

This problem gave me fits when I was in grad school. Looking back at it now, it still escapes me. The problem is from Conway's Functions of One Complex Variable. I'm looking for a proof from basic ...
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1answer
24 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
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2answers
26 views

In dual numbers, what number is represented by the following matrix?

In dual numbers, what number is represented by the following matrix? \begin{pmatrix}0 & 0 \\1 & 0 \end{pmatrix}
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1answer
28 views

Two different results with contour integration

This is probably going to be a stupid question ( I don't feel great today) but I can't get around this problem. $$I = \int_\mathbb R \frac 1 {(3x-2i)^2} dx $$ I thought that using contour ...
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1answer
35 views

$|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$

Let $p(z)= \sum_{k=0}^n a_k z^k$ , $a_n \neq 0$ , be a polynomial of degree $n$ such that $|p(z)| \leq M$ for $|z| \leq 1$. Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$ This was an exam ...
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1answer
49 views

$ \int_\gamma \frac{1}{z\sin z}dz$ where $\gamma$ is the circle $|z| = 5$

My understanding is that if this integral exists in the real sense, i.e. real Riemann-wise, then I can apply the residue theorem. If not, I may use the Cauchy Principal Value, to obtain a value. To ...
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22 views

an example of functions which is essentialy bounded but not continuous in circle

Can you give me an example of a function which is essentially bounded but not continuous in the unit circle and bounded in the open unit ball?
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1answer
39 views

How can I prove that the zeroes of $f(z)=1+1/2^z$ have no real part?

I want to prove that the zeroes of the function $f(z)=1+1/2^{z}$ have no real part. Is the following correct? $f(z)=0$ so $2^{z} = -1$ and $-1=e^{i\pi}$ so $e^{i\pi} = e^{z\ln2}$ therefore $z= ...
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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 ...
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41 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 ...
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3answers
28 views

How to show that a complex function have a branch in a domain

I've given as homework to show that the function $$f(z)=\sqrt{\frac{z+1}{z-1}} $$ has a branch on $G = \mathbb C \backslash [-1,1] $. I'm having a hard time in finding the way to approach this kind ...
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24 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
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1answer
78 views

An entire and one-to-one function must be of the form AZ+B, A non-zero. How to rule out higher degree polynomials in z? [duplicate]

Show that if f is entire and one-to-one, then it must be of the form AZ+B, with A not equal to zero. I am editing my question, since there are duplicates on this forum to the question of why an ...
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28 views

Is restriction of $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection?

I have this question: Is the restriction of exp function $exp:\mathbb{C} \to \mathbb{C}^*$ to $A = \{ x+iy : x \in R, y \in ]1, 1+2\pi]\} \subset \mathbb{C}$ a bijection? Here's what I tried: ...
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1answer
18 views

Biholomorphic mapping of $\tan(z)$

I'm supposed to solve this question: Show that the function $\tan$ maps the vertical strip $-\frac{\pi}{4}<x<\frac{\pi}{4}$ biholomorphically to $\dot B(0,1)$ It is obvious that ...
8
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1answer
50 views

Is this contour continuously deformable into a circle?

As an exam question, we had to solve the integral of $\frac{1}{z}$ over the following contour: (The contour is a sequence of straights arcs joining -1, -$\frac{i}{2}$, $\frac{1}{2}$, i, ...
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2answers
47 views

How to compute the residue of $(z^2+2z+1)\sin\left(\frac{1}{1+z}\right)$

This was an example given in my notes but all it concluded was with something about an infinite principal part. How do we compute it? we have it equal to $ \left( z + 1 \right)^2 \cdot \sin \left( ...
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1answer
16 views

The Laurent series expansion of 1/((z^2)(z+1))

In an example in our notes it says: Compute the Laurent series for f(z)= 1/z^2(z+1) and determine the annulus of convergence. No more information was provided. So I did it on my own by factoring it ...
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2answers
47 views

Integral of rational function in the complex plane

Let $P$, and $Q$ be complex polynomials such that $\deg Q \ge \deg(P) + 2$ Prove that there exists $r > 0$ such that if $\gamma$ is a closed curve outside $\{z : |z| \le r\}$, then $$\int ...
8
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1answer
84 views

Prove that an analytic function, real-valued on radii $[0, 1)$ and $[0, e^{i\pi\sqrt 2})$, is constant on the open unit disk

Suppose $f$ is analytic in the open unit disc and real valued on the radii $[0, 1)$ and $[0, e^{i \pi \sqrt 2})$. Prove that $f$ is constant. I'm not sure how to solve this.