# Questions tagged [complex-analysis]

For questions mainly about theory of complex analytic/holomorphic functions of one complex variable. Use [tag:complex-numbers] instead for questions about complex numbers. Use [tag:several-complex-variables] instead for questions about holomorphic functions of more than one complex variables.

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### If $f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$ has radius convergence $R > 0$. if $f(z) = 0$ for all $z$ $|z-z_o| < R$ show that $a_0 = a_1 = ... =0$.

If $f(z) = \sum_{n=0}^{\infty} a_n(z-z_0)^n$ has a radius of convergence $R > 0$ and if $f(z) = 0$ for all $z$ $|z-z_o| < R$ show that $a_0 = a_1 = ... =0$. Proof Attempt: If it has a radius of ...
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### Does it mean $\displaystyle\lim_{z=0}\oint\frac{d}{dz}\frac{z}{f(z)^n}dz=0$?

Source of my question My question comes from the book "Introduction to Analysis" (by 高木貞治)  → Chapter 7 "Sequel to Differentiation (Implicit Functions)"  → Section 85 "...
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### Why is it the case that $|z-\alpha|\leq n\left|\frac{p(z)}{p'(z)}\right|$ for a polynomial $p$ of degree $n$ with root $\alpha$?

I was reading this article which claimed the following: $$|z-\alpha|\leq n\left|\frac{p(z)}{p'(z)}\right|$$ Where $p$ is a polynomial with degree $n$, and $\alpha$ is the zero of $p$ closest to $z$. ...
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### $\sum{z^n\over (n!)^\alpha}$ is an entire function of order $1/\alpha$ [duplicate]

The problem is from Stein Complex analysis Chapter 5 Problem 3. Show that $\sum {z^n\over (n!)^\alpha}$ is an entire function of order $1/\alpha$. The problem was already posted here, but it seems ...
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### References regards complex analysis on Banach spaces

I'm looking for a references on complex analysis in Banach spaces. What are good books on this subject? Any help is welcome. Thanks in advance.
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### showing the image of unit disc on Complex plane under $f(z)=z^2$ is open

I want to use the fact that $|f(z)|=|z|^2$ and $|z|<1$(so basically the map f maps onto some subset of the unit disc that includes {-1,1,i,-i}) to construct an open disc for any arbitrary point in ...
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### Show that $\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$ is meromorphic

Let $p\geq 1$ be an integer. If $\sum_{n=1}^\infty 1/|a_n|^{p+1}$ converges, show that $$\sum_{n=1}^\infty\left({1\over z-a_n}+{1\over a_n}+{z\over a_n^2}+\cdots+{z^{p-1}\over a_n^p}\right)$$ is ...
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### Integral formula for a complex function

I am working through some problems in complex analysis, and am stumped by the following: Let $f$ be an analytic function in the disk $D=\{z\in\mathbb{C}:|z|<R\}$ and $\gamma$ be the ...
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### Calculate $\int_{\gamma}\cot z dz$ where $\gamma=C(0,3)$

I've just started doing complex integrals with residues, and I'm struggling a bit with finding the singularities. The first integral is: $$\int_{\gamma}\cot z dz$$ where $\gamma=C(0,3)$. My solution ...
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### Find an example of unbounded subset of complex set.

Find an example of unbounded subset of complex set. Attempt: The set $A=\{e^{z}: z \in \Bbb C\} \subseteq \Bbb C$ is unbounded over $\Bbb C$. Proof Let $M>0$ be an arbitrary real number. Then there ...
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### Rudin's RCA: $2.14$ theorem step $X$

This is the definition which we need in the proof: source : Rudin proof of theorem 2.14 RCA inequality explanation Proof. Clearly, it is enough to prove this for real $f$. Also, it is enough to ...
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### Derive solution to $\int_{0}^{+\infty }\left( \frac{1}{1+s^{2}}\right) ^{\nu }ds$ where $\nu$ is noninteger real

Can anyone derive the solution to $$\int_{0}^{+\infty }\left( \frac{1}{1+s^{2}}\right) ^{\nu }ds$$ where $\nu$ is a non-integer positive real? The derivation has been provided under this and this ...
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### Obtaining a uniform expression for a family of inverses of perturbations of the identity

Let $h$ be a $C^2$ function with compact support. Define the family of functions $h_t(x) = x + th(x)$. They look like pertubations of the identity. For small $t$ we can show that $h_t$ is a $C^2$ ...
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### The Riesz representation theorem Rudin's RCA book: step $X$

This is the definition which we need for the theorem: There is the theorem: Let $X$ be a locally compact Hausdorff space, and let $\Lambda$ be a positive linear functional on $C_{c}(X)$. Then there ...
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### Intuition behind Laurent series expansion

I know Laurent series has to do something when function has singularities.I am trying to get idea why Laurent series in such way.. For that i read at somewhere... "A taylor series requires the ...
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### How do we evaluate the inverse laplace transform?

Here's two answered and upvoted questions that do not mention anything about a region of convergence in their premises. Usage of inverse Laplace transform Inverse Laplace Transform properities It is ...
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### Looking for interesting math functions that alternate between multiple different trigonometric parameters

I am looking for functions that do something like f(x) = { log(x) , if x is a multiple of 3; sin(x) , if x is not a multiple of 3 } But not being super random like the above (preferrably somewhat ...
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### Finding the Laurent series of the function $f(z) = \frac{1 - 2z}{(z^2 - z)^2}$ on the annulus $\Delta^*(0,1)$ (i.e. $0 < |z| < 1$)

I am tasked to find the Laurent series of the function $f(z) = \frac{1 - 2z}{(z^2 - z)^2}$ on the annulus $\Delta^*(0,1) = \{z \in \mathbb{C}\mid 0 < |z| < 1\}$. The hint I have been given is to ...
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### Finding poles or removable singularities of $f(z)=z\cot(z)$

Im trying to find pole or removable singularities, and also determining their order of following function: $f(z)=z\cot(z)$ My solution so far is: $f(z)=z\cot(z)=z\frac{\cos z}{\sin z}$ $\Rightarrow$ ...
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### How to invert linear operators with functional coefficients?

So in this episode we described a way to invert linear operators with constant coefficients, that is operators of the form $O[f] = c_0f + c_1 f' + c_2 f'' ... = \sum_{n=0}^{\infty} c_n f^{(n)}$. Now ...
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### How to invert a linear operator with constant coefficients?

Given a linear operator $O[f] = c_0 f + c_1 f' + c_2 f'' + c_3 f''' ... = \sum_{n=0}^{\infty} c_n f^{(n)}$, where all the $c_i$ are constant is it possible to find a nice closed form for the inverse ...
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