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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31 views

If $P$ is a polynomial of degree $n>0$, then there exists circle $C$ of radius $R$ such that $\int_{C} \frac{P^{\prime}(z)}{P(z)} dz=2n\pi i $

Let $P$ be a polynomial of degree $n>0.$ Could anyone advise me on how to show there exists $R>0$ such that if $C$ is the circle $|z|=R$ anticlockwise oriented, then $\begin{align}\int_{C} ...
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42 views

complex limits, how to show they go to 0?

In complex integration my book uses that some limits go to zero as R goes to infinity. However I do not now how to show this, these two limits are: $e^{-\pi(R^2+2iRy-y^2)}$, where y is a real number ...
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1answer
16 views

Parallel complex vectors

Show that vectors defined by complex numbers $z_1$ and $z_2$ are parallel if and only if $\operatorname{Im}(z_1 \bar{z_2})=0$. Does the proofing use the fact that in $\mathbb{R}^2$ the cross product ...
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0answers
35 views

complex analysis - differentiabiliity

The exact question is as follows: Determine all points in the complex plane where the following function is differentiable. $$ f(z)=\frac{1}{z^2+iz+1}+\cosh(\sin(z)) $$ I'm going to use the Cauchy ...
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2answers
37 views

Complex analysis: Rewrite $\cos^{-1}{i}$ in algebraic form

I'm stuck in this problem (complex analysis), my answer is not the one reported in the book: Rewrite $\cos^{-1}{i}$ in the algebraic form. A: $k\pi + i \frac{\ln{2}}{2}\ \forall\ k \in \mathbb{Z}$ ...
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1answer
44 views

Determining a power series of $f(z)=\exp(z^{2})$

Let $f(z)=\exp(z^{2})$. Determine a power series of $f$, i.e the coefficients $a_{k}$ such that $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ for all $z\in \mathbb{C}$, where $a_{k}=f^{(k)}(0)/k!$. First ...
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0answers
38 views

Prove the complex function is entire

The function $$ f(z) = e^{x^2 - y^2} (\cos 2xy + i \sin 2xy )=e^{z^2} $$ Then, how to prove it's analytic everywhere of complex plane of the exp function...?
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2answers
37 views

Does the analytic continuation of $f$ always exist?

Let $f(z)$ be a holomorphic funtion on region $\Omega$. Then, does the analytic continuation of $f$ always exist? Note that $f$ is always the analytic continuation of itself, so I exclude this case. ...
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1answer
27 views

Showing that a function is not meromorphic on $\mathbb{C}$

Please, I need some advices to solve this exercise: Let $f$ be meromorphic on $\mathbb{C}$ but no entire. Let $g(z)=e^{f(z)}$. Show that $g$ is not meromorphic on $\mathbb{C}$. I appreciate your ...
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1answer
42 views

Is there anything special with complex fraction $\left|\frac{z-a}{1-\bar{a}{z}}\right|$?

Is there anything special with the form: $$\left|\frac{z-a}{1-\bar{a}{z}}\right|$$ ? With $a$ and $z$ are complex numbers. In fact, I saw it in a problem: If $|z| = 1$, prove that ...
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1answer
156 views

How to solve this system of differential equations in a complex variable?

I would like to solve the system: $$ \begin{array}{rclr} \frac{d}{dz} u & = & \frac{\alpha}{z^k} u & + v \\ \frac{d}{dz} v & = & & \frac{\alpha}{z^k} v \end{array} $$ where ...
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1answer
79 views

$\log(z_1\cdot z_2\cdot \ldots \cdot z_n)=\log(z_1)+\log(z_2)+\ldots+\log(z_n)$

I came across this question: Let $z_1, z_2, \cdots , z_n$ be complex numbers What conditions must be met so that $$\log(z_1\cdot z_2\cdot \ldots \cdot z_n)=\log(z_1)+\log(z_2)+\ldots+\log(z_n)$$ ...
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24 views

The integral of a monomial over the complex sphere

Let $\alpha=(\alpha_{1},...,\alpha_{q})\in\mathbb{N}^{q}$ a multi-index. What is the expression for $$\int_{S}z^{\alpha}\,d\sigma(z),$$ where $S$ is the unit sphere of $\mathbb{C}^{q}$ and $\sigma$ ...
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2answers
32 views

Simplify $w=\frac{(1+i)z-i+1}{iz-1}$

I have difficulties understanding how this expression $$w=\frac{(1+i)z-i+1}{iz-1}$$ is simplified to this $$w=1-i-2\cdot\frac{1+i}{z+i}$$ Here are some steps from my exercise notebook: ...
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0answers
37 views

Analytic Continuation of the zeta function

Is the analytic continuation of the Riemann zeta function to the upper half plane unique? I don't know much complex analysis, so I can't see why that is the case.
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1answer
25 views

Stereographic projection is conformal — from the line element

I'm looking over some fairly basic stuff on complex methods and the book I'm using takes the formula for the stereographic projection: $$z = \cot(\beta/2)e^{i\phi} $$ as well as the line element on ...
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0answers
12 views

Behavior of lower incomplete gamma function at complex infinity

The lower incomplete gamma function is given by $ \gamma\left(s, x\right) = \int\limits_0^x t^{s-1} e^{-t} {\rm d} t~,$ and has a well-defined analytic continuation for both $s$ and $x$ [1]. ...
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0answers
32 views

Methods for “recognizing” a polynomial of several variables.

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial. 1) Show that $f^{(n)}\equiv0$ for some $n\geq1$. 2) ...
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3answers
40 views

Proof for argument identity

I have trouble algebraically show proof for this well known statement: $$-\operatorname{arg}(z)=\operatorname{arg}(z^{-1})=\operatorname{arg}(\bar{z})$$ if $z=x+yi$ and ...
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3answers
61 views

Continuity of conjugate of $z$: $f(z)=\bar z$

How to prove that the function $\;f(z)=\bar z\;$ is continuous on the whole plane?
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26 views

Boundedness of sequence

Consider a dynamical systems over complex numbers $$ z_{n+1}=\frac{\alpha}{z_{n}}+ \frac{\beta}{z_{n-1}},\qquad n=0,1,\ldots $$ where the parameters $\alpha, ~\beta$ are complex numbers, and the ...
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27 views

Interchange of infinite product and limit

The Problem Let $(a_{n,m})_{n,m\in \mathbb{N}}$ be an sequence of complex numbers. Under which conditions can I interchange product and limit? $\lim_{m\to\infty}\prod_{n=1}^{\infty} ...
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27 views

How to show that a point is a branch

If a function $f(z)$ has a branch point around $\alpha$ than the endpoints of a path around $\alpha$ do not map to the same point under $f(z)$. So we must test for inequality of $f(\alpha + ...
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25 views

Orientation of circles in $\Bbb C_\infty$

Let $\Gamma$ be arbitrary and suppose $z_1,z_2,z_3 \in \Gamma$, then for any mobius transformation $S$ we have, $\{z:Im(z,z_1,z_2,z_3)\gt 0 \}$...........$(a)$ = $\{z:Im(Sz,Sz_1,Sz_2,Sz_3)\gt 0 ...
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31 views

complex numbers in binomial function form

Is it possible to rewrite $f(z) = z^{n}(z^*)^{m}$ with the use of the binomial function. The binomial function is defined as $(x+y)^{n} = \sum\limits_{k=0}^n \binom{n}{k} x^{n-k}y^{k} = ...
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0answers
29 views

Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point

Proof that f(z) = $z^{n}(z^*)^{m}$ is not analytic in any point. If i look at the limit of a more simple function of this form: f(z) = $\frac{z}{z^*}$ I would say that the limit does not exist, ...
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1answer
38 views

Few questions on complex number

Hey few statements (correct / not correct) I'm not sure about. If $z_1$ and $z_2$ are complex numbers so that $z_1 + z_2$ and $z_1 z_2$ is a negative real number then $z_1$ and $z_2$ are real ...
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5answers
176 views

Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$

Do you know any nice way of expressing $$\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$$ ? Some simple manipulations involving the integrals lead to an expression that also uses the hypergeometric series. ...
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1answer
22 views

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$

Describe the set of all harmonic functions $u(x,y)$ in $\mathbb{C}$ such that the product $(x^2 −y^2)u(x,y)$ is harmonic in $\mathbb{C}.$ I have concluded that $xu_x = yu_y$. Not sure how to proceed ...
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21 views

When are two complex numbers symmetric with respect to a straight line

If $\Gamma$ is a circle through points $z_2,z_3,z_4$ then $z^*$ and $z$ in $\Bbb C_\infty$ are said to be symmetric if $(z^*,z_2,z_3,z_4)$= Conjugate of $(z,z_2,z_3,z_4)$ (I couldn't get an overline ...
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0answers
34 views

$√2|z|≥|Rez|+|Imz|$ [duplicate]

I just came across this simple inequality which I am finding it difficult to prove. For any complex number z I need to prove that the following inequality holds $$√2|z|≥|Rez|+|Imz|$$ I would much ...
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1answer
50 views

Application of Opening Mapping theorem

Let $f$ be a holomorphic function on open set $A$ such that $(Im(f(z))^3 + (Re(f(z))^4 =5.$ Could anyone advise me on how to use Open mapping theorem to prove $f$ is constant? Hints will suffice. ...
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2answers
39 views

Condition for zeros of a polynomial in Unit Disk

Consider a polynomial in $\mathbb{C}$ with complex coefficients, $\lambda^2+p\lambda+q$ where both $p$ and $q$ are complex numbers. I am looking a for a condition of $p$ and $q$ such that the zeros ...
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1answer
54 views

Suppose that $a_0 >a_1 >…>a_{2013} >0.$ Prove that $\sum_{n = 0}^{2013}a_nz^n \neq 0$ when $|z|<1$ [duplicate]

Suppose that $a_0 >a_1 >...>a_{2013} >0.$ Prove that $\sum_{n = 0}^{2013}a_nz^n \neq 0$ when $|z|<1$ Not sure where to begin with this. Any suggestions? Thanks.
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2answers
41 views

Finding $\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$

$\large\zeta_7\left(\zeta_3\right)^5$ where $\large\zeta_n=\cos{\frac{2\pi}{n}}+i\sin{\frac{2\pi}{n}}$ I am having trouble getting a final answer that makes sense to me. Here is what I tried: ...
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0answers
29 views

Prove that every complex number is in the range of the entire function $e^{3z} + e^{2z}.$

Prove that every complex number is in the range of the entire function $e^{3z} + e^{2z}.$ By Picard we have that every number except maybe one is, but that is all I've got. Help would be great! ...
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2answers
45 views

How to integrate $x\times \frac{\sin(x)}{x^2+a^2}$ from zero to infinity

I am trying to evaluate $\int_0^\infty\frac{x \sin(x)}{x^2+a^2} dx$. I get $\frac{\pi}{4} \sin(ia)$ using residue theorem. I integrated over the path that goes from -R to R along the real axis and ...
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1answer
29 views

Prove that $f$ is the restriction to the unit disk of some entire function, given $f(2z)=2f(z)f'(z)$

Suppose $f$ is holomorphic in the unit disk, and $f(2z)=2f(z)f'(z)$ whenever $|z| < 1/2$. Prove that $f$ is the restriction to the unit disk of some entire function. Not sure how to attack this ...
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0answers
29 views

Complex Integrals using a contour [duplicate]

Can anyone help me how to prove this integral
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0answers
20 views

A question from Needham's Visual Complex Analysis.

I am trying to understand an excerpt from Needham's Visual Complex Analysis. This is from pg. 352. If $p$ lies in the darkly shaded outer region, then shouldn't $v[\Gamma,p]=1$?
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4answers
175 views

Can I conjugate a complex number : $\sqrt{a+ib}$?

Can I conjugate a complex number: $\sqrt{a+ib}$ ? Actually my maths school teacher says and argues with each and every student that we can't conjugate $\sqrt{a+ib}$ to $\sqrt{a-ib}$ because ...
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1answer
35 views

complex integration, how to evaluate it?

I have this exercise: Show that if $|a| < r <|b|$, then $\int_\gamma \! \frac{1}{(z-a)(z-b)} \, \mathrm{d}z=\frac{2\pi i}{a-b}$, where $\gamma$ denotes the circle centered at the origin, ...
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1answer
50 views

holomorphic function on $R=\{z\in \mathbb{C} \mid |z|<1\}$

Let $A$ be a set of all injective and holomorphic functions on $R=\{z\in \mathbb{C} \mid |z|<1\}$ with $f(0)=0,f'(0)=1$ then (1) for any $a\in R$ prove that function $g(z)=\frac{z}{(1-az)^2}$ is ...
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2answers
20 views

Do these functions have an holomorphic extension to $0$?

I want to know if the following functions have an holomprohic extension to $0$ or not: (a) $\frac{z}{e^z-1}$. (b) $z^2 \sin \frac{1}{z}$. For (b) I think it does because for $0<|z|<1$, $|z^2 ...
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0answers
49 views

Suggested book for self study.

I have a degree in Financial Risk Management, and did 4 semesters of calculus and analysis(but that was about 10 years back), with most of my other efforts going toward Mathematical Statistics and ...
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0answers
46 views

Modulus of a complex function

Suppose $\alpha$ and $\beta$ are two arbitrary complex numbers. Let $$f_{\pm}(\alpha, \beta)=\frac{\frac{-\alpha}{\alpha+\beta}\pm ...
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1answer
39 views

A few questions about the winding number.

Let $C_a$ be a circle in the anti-clockwise direction around point $a$ in $\Bbb{C}$. If $f:\Bbb{C}\to\Bbb{C}$ is a continuous mapping, then $f(C_a)$ is the image of $C_a$ in $\Bbb{C}$. I have a few ...
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1answer
65 views

Solving the equation of damped oscillator

I'm asked to prove that any solution of the equation $$\ddot\Phi+\Gamma\dot\Phi+\omega_0^2\Phi=0;\qquad \omega_0>\frac\Gamma 2$$ is $$\Phi=A_0e^{-\frac{\Gamma}{2} t}e^{i(\omega t-\beta)};\qquad ...
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2answers
39 views

Laurent Series on the boundary

Suppose $f(z)$ is analytic for $|z| \geq 1$, then it is analytic in $|z| >1$ and therefore has a Laurent series $f(z) = \sum_{n=-\infty}^\infty c_n z^n,~\forall |z| > 1$. Is it also true that ...
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0answers
27 views

Boundary of a holomorphic functions

Let $G ⊆ \mathbb{C}$ a bounded open connected set and let $f : \bar{G} → C$ a holomorphic function: Is this true? $$∂f(G) ⊆ f(∂G)$$ What I have is that $f$ is open.