# Questions tagged [analytic-functions]

For questions about analytic functions, which are real or complex functions locally given by a convergent power series.

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### How is this argument in this paper justifiable?

Eq. 10 in this paper https://arxiv.org/pdf/1806.04039.pdf has an integral equation \begin{equation} \sigma(\lambda)=a_1(\lambda)+\frac{1}{N}a_1\left(\lambda+\frac{1}{g}\right)-\int_{\mathcal C} d\...
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### Show that $f(z)$ is constant [duplicate]

Suppose that $f(z)$ is entire with $a$ and $b$ positive constants. If $$f(z+a)=f(z+b i)=f(z)$$ for all $z$, show that $f(z)$ is constant . Which theorem could I use ? Please give some hint thankyou
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### Cauchy integral formula for a point outside the simple closed curve

Show that if $f$ is analytic inside and on a simple closed curve $C$ and $z_0$ not in $C$ then $$(n-1)!\int_C \frac{f^{(m)}(z)}{(z-z_0)^n}dz=(m+n-1)!\int_C \frac{f(z)}{(z-z_0)^{m+n}}dz$$ Since it is ...
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### Analytic image of conjugate of a point

Let $f:C \rightarrow \{|z|\leq2021 \}$ be an analytic function. If $f(11+9i)=\frac{1}{2i}$, then find the value of $f(11-9i)$. How should I proceed with the problem ?
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### Give the points at which the given functions will not be analytic [closed]

I don't know how to prove that a function is not analytic at a certain point help out, please $$f(z) = \frac z {z - 3i}$$ Basically prove that it is not analytic at a "certain point"
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### How to extend analytic function

Let $\gamma:\left(a,b\right)\to\mathbb{C}$ be an analytic path (Meaning, near every $t_{0}\in\left(a,b\right)$ the function $\gamma$ has a Taylor expansion in $t-t_0$ with positive radius of ...
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### Prove that there doesn't exists a holomorphic function defined in the disk $D(0,2)$ which satisfies the following condition

Prove that there doesn't exist a holomorphic function defined in the disk $D(0,2)$ which satisfies $$\frac{(-1)^n}{n}+e^{f(1/n)}=1 \space \forall n\in \mathbb{N}.$$ I am having trouble proving this. I ...
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### If $f(z)$ is analytic then $f(\bar z)$ is analytic iff $f$ is constant. [duplicate]

I need to prove this using the C-R equations. I am a beginner and I really don't know much about the methods of proof usually employed in such questions. Please help. Thanks in advance.
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### Understanding Identity theorem proof in complex analysis

Identity theorem Let $f$ be holomorphic functions on some domain $D\subset\mathbb{C}$, and let $S$ be the set of all zeros of $f$ which has a limit point in $D$. Then $f$ is identically zero in $D$. ...
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### Analytic function with arbitrary, countable zero set

Let $S\subset\mathbb{C}$ be a countable set with no accumulation points. Does there exist an analytic function $f(z)$ such that $f(z_i)=0$ for all $z_i\in S$ and $f(z)\neq 0$ for $z\neq S$? Work so ...
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### Zeros of analytic function is at most countable [duplicate]

Is this statement true? Each analytic function on an open set can have at most countable number of zeros. Can you give a me proof please? Thank your sir and Madam.
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### Hilbert transform of x (or a sawtooth)

This has a simple curiosity value, but I was converting the czt() (chirp z-transform) function from Octave into (wx)Maxima and, while testing it, I used a simple ...
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### Is the zero set of an analytic function closed nowhere dense?

Given a non-constant analytic function $f(x)$ on a domain $D \subseteq \mathbb{R}^n$. I want to prove that $\mathcal{Z} = \{x \in \mathcal{D} |f(x) = 0 \}$ is closed nowhere dense. I originally wanted ...
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### Specific branch of complex logarithm

Find a branch of $\log(3z-2)$ which is analytic at $z=0$ and takes the value $\ln 2+\pi i$ there. Specify the domain on which this branch is analytic. These are my thoughts. I know that $\log z$ is ...
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### Ignoring higher order terms in computations involving real analytic functions

We have the following four analytic functions of a real variable: \begin{align} \lambda (\epsilon) &= 0 + \lambda_1 \epsilon + \lambda_2 \epsilon^2 + \ldots \\ \tilde{\lambda} (\epsilon) &= 0 ...
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### Prove or disprove a proposition on Taylor series

Let $f(x) = \sum_{k\geq 0} a_kx^k$ be a real analytic function, converging on $-R < x < R$, whose $n^{th}$ Taylor polynomial is $f_n(x) = \sum_{k=0}^n a_kx^k$. If there exist $N\in\mathbb{N}$ ...
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### The Zeta Function and the Prime Number Theorem

I an trying to learn more about the $\zeta$ function, and I stumbled upon this very interesting paper https://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Riffer-Reinert.pdf and having a few ...
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### Convergence of power series at the boundary

Consider the following complex power series $$\sum_{n\geq 1}{\frac{ni^n}{2^n}{z^{n-1}}}$$ By the root test, I have concluded that the disc of convergence is $D:=D(0,2)$. Then, I would like to study ...
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### Fixed Points in Mobius Transformation

Let $$w(z)=\frac{e^z-1}{e^z+1}; \ z=x+iy$$ and $$w=u+iv$$ Then show that the image of the $y$-axis in the domain is $v$-axis in the co-domain. Does this contradict the result that a Mobius ...
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### A question regarding differentiability and the boundary of analytic functions

Consider a Jordan curve $\Gamma\subset \mathbb{C}$ and let $\Omega$ be the interior of $\Gamma$ (which is well-defined by the Jordan curve theorem). Let $f:\Gamma\cup\Omega\rightarrow\mathbb{C}$ be ...
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### Prime ideals in ring of real analytic maps correspond to closed submanifolds?

In affine algebraic geometry, prime ideals in a ring $R$ correspond to irreducible subschemes of $Spec(R)$. For a real, compact, smooth manifold $M$, each prime ideal in $\mathcal{C}^\infty(M)$ is ...
### Is the complex function $f(z) = \sec z$ analytic?
Please help. Is $f(z) = \sec(z)$ analytic? I do know that I have to test if the cauchy riemann equations hold. That is $U_x = V_y$ and $U_y = -V_x$ But I have trouble expressing $f(z)$ in the form \$f(...