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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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An understanding problem of analytic function

I am reading a textbook of distribution theory. Here is the definition of test functions: The class of test functions $\mathcal D(\Omega)$ consists of all functions $\varphi(x)$ defined in $\Omega$, ...
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19 views

Can the following equation be solved analytically?

Is it possible to solve the following equation analytically for r: $$A = exp\bigg[a\bigg(\frac{r}{r_{0a}}\bigg)^b\bigg] + exp\bigg[c\bigg(\frac{r}{r_{0c}}\bigg)^d\bigg] + exp\bigg[f\bigg(\frac{r}{r_{...
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2answers
21 views

Does this equation have any analytical solution?

I am trying to compare my numerically calculated model with analytical solution, which should be provided by following equation $f(x) = e^{\frac{[f(x)]^2}{2}+ax+b}$ where $a$ and $b$ are constant. ...
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2answers
83 views

Find the zeros of $f(z)=z^3-\sin^3z$

I want to find the zeros of $f(z)$, $$f(z)=z^3-\sin^3z$$ My attempt $f(z)=0$ $z^3-(z-z^3/3!+z^5/5!-\dots)^3=0$ $z^3-z^3(1-z^2/3!+z^4/5!-\dots)^3=0$ $z^3[1-(1-z^2/3!+z^4/5!-\dots)^3]=0$ So $z=0$ ...
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1answer
20 views

Fixed points of complex analytic functions

We have given some complex analytic function $f:\mathbb{C} \to \mathbb{C}$. I have read in a proof that if $f$ has exactly one fixed point $v \in \mathbb{C}$, then follows that $f$ is a polynomial ...
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1answer
20 views

Existence of convergent sequence in domain

Im looking at a proof where there are two analytic functions $f,g$ where $g$ is injective and maps a domain onto the open unit disk $\mathbb{D}$ and $f$ maps the same domain onto a domain $G$ with $\...
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3answers
49 views

Analytic function is identically zero on open unit disk if $|f(z)|\leq 1-|z|$. [duplicate]

Let $f$ be an analytic function defined on open unit disk with $|f(z)|\leq 1-|z|$. I need to establish $f$ is zero on disk . If I'm able to prove that $f(0)=0$ and $f(0)\leq f(z)$ then $f$ is ...
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1answer
32 views

If $f(z)$ is analytic inside and on the circle $|z| = 2$, is $\oint_{|z|=2}f(z) = 0?$

I know if $f$ is analytic in a simply connected domain D then for any closed loop in D this integral is 0. Is that enough to say yes to this question?
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1answer
22 views

Question about proof that $Ln$ is analytic.

I have a question about a step in the proof of the following theorem. Theorem: $Ln: \mathbb{C}- \{x \in \mathbb{R}: x \leq 0\} \to \mathbb{C}: z = re^{i \phi} \mapsto \ln r + i \phi$ $(\phi \in (-\pi,...
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Contour Deformation in the Laplace Inversion Formula

Following Szpankowski - Average Case Analysis of Algorithms on Sequences the exponential generating function of $g(n)$ which is thought to be analytic in $n$ is defined as $$ G(z)=\sum_{n=0}^{\infty} ...
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2answers
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Prove that there exists no non-zero analytic function such that $f(\frac{1}{n})$=0 for all n

There exists no non-zero analytic function such that $f(\frac{1}{n})$=0 for all n Im trying to prove by differentiability If $f$ is differentiable it is continuous at $0$, So $f(0) = \lim f(\frac{1}...
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1answer
36 views

A query while showing that the Gamma function $\Gamma$ is logarithmically convex for $x \gt 0.$

We are using the general definition of gamma function defined on $\Bbb C \setminus \{\text{non-positive integers}\}$ i.e. $\Gamma(Z)=\frac {e^{-\gamma z}}{z} \prod (1+ \frac zn)^{-1} e^{\frac zn}$. ...
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1answer
65 views

Is there any difference between the elements in $\mathbb{C}[x]$ and $C^{\omega}$?

Studying Hilbert spaces, I've been told that $\{1, x, x^2,\dots, x^k,\dots\}$ form a basis (EDIT: when orthonormalised) in $\mathcal{L}^2([a,b])$ because the set of all polynomial functions is dense ...
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1answer
33 views

$f:D(0,1) \to \mathbb{C}$ holomorphic injective such that $f(0) = 1 = f'(0)$ and $f^{(k)}(0) \in \mathbb{R} \; \forall k$

Let $f:D(0,1) \to \mathbb{C}$ holomorphic injective such that $f(0) = 1 = f'(0)$ and $f^{(k)}(0) \in \mathbb{R} \; \forall k$. Show that if $\Omega^+ = \{z \in D(0,1):im(z)\geq 0\}$ and $\Omega^-...
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27 views

Is there a term for extensions of functions on discrete sets to continuous sets?

The pi function, $\Pi(z)$ – defined as $\Pi(z) = \Gamma(z+1)$, where $\Gamma(z)$ is the gamma function – extends the factorial in that $$\Pi(n) = (n)!$$ for all positive integers $n$. In other words,...
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1answer
28 views

Confusion about the definition of analytic and singularity.

In my textbook the definition of analyticity is given as A function is said to be analytic in a domain D if f(z) is defined and differentiable at all points of D. The function f(z) is said to be ...
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17 views

Each square integrable harmonic function can be written as the sum of an antianalytic and an analytic, square integrable functions.

I want to prove that each square integrable harmonic function with respect to the standard Gaussian measure: $\gamma ^{{n}}({\mathbb{C}})={\frac {1}{{\sqrt {2\pi }}^{{n}}}}\int _{{\mathbb{C}}}\exp \...
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1answer
58 views

If $f,g:\mathbb R \rightarrow \mathbb R$ are analytic and $f=g$ on an interval of positive length, can we conclude that $f=g$ everywhere?

If $f,g:\mathbb R \rightarrow \mathbb R$ are analytic and $f=g$ on an interval of positive length, can we conclude that $f=g$ everywhere? I guess it is more like a theorem than a problem. I am ...
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Are conformal maps between Riemannian manifolds real-analytic?

Let $M,N$ be oriented smooth ($C^{\infty}$) $n$-dimensional Riemannian manifolds, and let $f:M \to N$ be a smooth orientation-preserving weakly* conformal map. Do there exist real-analytic ...
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Prove $f(z)=c$ when $\operatorname{Re}(f)$ has an upper bound ($f$ is analytic) [duplicate]

As the title says Prove $f(z)=c$ when $\operatorname{Re}(f)$ has an upper bound, given that $f$ is analytic I am just getting introduced on complex analysis, and I found proving that $f(z)=c$ ...
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26 views

Extending the domain of an analytic function to a larger set

I have the following problem: Given some function $f : \mathbb C \rightarrow \mathbb C$ that is analytic on some vertical line segment, there exists some (non-trivial) rectangle around that line ...
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1answer
154 views

Milne Thomson method for determining an analytic function from its real part

What is the logic behind taking $z = {\bar {z}}$ while finding analytic function in Milne-Thomson Method ? I mean we write $ { f(z)=u(x,y)+iv(x,y)} $ as $ {\displaystyle f(z)=u\left({\frac {z+{...
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1answer
79 views

Are constant and identity only analytic functions such that $f(f(x))=f(x)$?

If a function over reals is given by $$f(x) = \sum_{n=0}^\infty a_n x^n$$ and satisfies $$f(f(x))=f(x),$$ does this imply that $f(x) = x$ and $f(x) = c$ are only valid choices for $f(x)$? It seems ...
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2answers
54 views

The Weierstrass theorem from complex analysis

The Weierstrass theorem from complex analysis states the following: Suppose $f_n$ is a sequence of analytic functions converging uniformly on an any compact subset of its domain to $f$. Then $f$ is ...
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2answers
67 views

Using contour integration for the inverse Laplace transform to find the inverse transform of $\dfrac{s}{s^2 + a^2}$

I am trying to use the contour integration formula for the inverse Laplace transform, find the inverse transform of $\dfrac{s}{s^2 + a^2}$. My textbook says that the solution is $\cos(at)$, but it ...
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1answer
35 views

How to show a real valued function of several variables is analytic?

Let $f:\Omega\subset\Bbb{R}^m\to\Bbb{R}^n$ be a given function. In general, how can I show that $f$ is smooth (infinitely differentiable) and analytic. I know this is a bit vaguely stated question, ...
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Associative, non-commutative, non-trivial, analytic binary operation

There was a question whether associative, but non-commutative binary operation over the real numbers exist. A trivial answer is the binary operation $x\circ y = x$ or $x\circ y = y$. As a followup, ...
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1answer
31 views

Show that a complex Differentiable function f with $|f'|\leq 1$ is a contraction.

I am working on the proof of the following statement; Suppose $f$ is analytic on a a rectangle $R$ and $|f'(z)|\leq 1$ for all $z \in R$. Then $f$ is a contraction on $R$, that is $$ |f(b)-f(a)|\leq |...
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1answer
37 views

Are two real, two variable polynomials, satisfying the Cauchy-Riemann equations, a complex polynomial?

Let $u, v \in \mathbb{R}[x,y]$ satisfying $u_{x} = v_{y}$ and $u_{y} = -v_{x}$ everywhere in $\mathbb{C}$. Is the function $f(x + iy) = u(x,y) + iv(x,y)$ a polynomial in the variable $z = x + iy$? I ...
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1answer
45 views

Analytic and bounded implies uniform continuity

Let $f$ be analytic and bounded in $\{z\in\mathbb{C}\mid Re(z)>0\}$. Prove that $f$ is uniformly continuous in $\{z\in\mathbb{C}\mid Re(z)>C\}=:D$ for every $C>0.$ For uniform continuity, I ...
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1answer
57 views

Show that a function with specific bound on its derivatives is analytic

I'm solving old exam problems in real analysis. Thus, only such methods may be used. I've been trying to solve the problem below and have encountered some issues. Let $f\in C^\infty(\mathbb{R})$ in ...
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1answer
49 views

Branch Points of the Polylog function

The polylogarithm $$ {\rm Li}_s(z) = \sum_{n=1}^\infty \frac{z^n}{n^s} $$ has obvious branch points at $z=1$. For integers $s\leq 0$ it is a rational function with a pole of order $1-s$ at $z=1$. If $...
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2answers
132 views

sum expression in terms of special functions $\sum_{n=1}^\infty \frac{n^{s-1}}{e^{2\pi n}-1}$

As the title already says it I have this expression $$ f(s)=\sum_{n=1}^\infty \frac{n^{s-1}}{e^{2\pi n}-1} $$ and am wondering if this one can be expressed in terms of any special or analytical ...
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1answer
43 views

Entire extension of $f(x+y)=g(x)g(y)-h(x)h(y)$

I am currently working on the following practice question for complex analysis; Assume $f(x+y)=g(x)g(y)-h(x)h(y)$ for all $x,y\in \mathbb{R}$ and some entire functions $g,h$. Show that there exists a ...
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1answer
253 views

How to argue that a series that isn't a power series is analytic?

Let $f(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{1-z^{n}}$. Then $f$ converges in the unit disc. I want to show that it is analytic in this region as well, but since it is not a power series I don't have ...
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1answer
72 views

If $f$ is entire and $\left|f\left(\frac{1}{\ln{(n+2)}}\right)\right|<\frac{1}{n}$ for every positive integer $n$ then $f=0$

Let $f(z)$ be an entire function satisfying $$\left|f\left(\frac{1}{\ln{(n+2)}}\right)\right|<\frac{1}{n}$$ for every $n\in\mathbb{N}).$ Show that $f(z)=0.$ I need some help for this ...
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1answer
34 views

Real differentiable function with sequence of turning points that tend to a limit… can the function be analytic?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be non-constant and differentiable with a sequence of $(x_n)_{n \in \mathbb{N}}$ such that $f'(x_n)=0 \, \forall \, n$ and $x_n \rightarrow c$ for some $c \in ...
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Is a real-analytic function which vanishes on a set of positive measure identically zero? [duplicate]

Suppose $n>1$, and let $U \subseteq \mathbb{R}^n$ be an open connected set. Let $f$ be a real-analytic function on $U$. Suppose that $f=0$ on a subset of $U$ of positive measure. Is it true that $...
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1answer
63 views

Showing $|1-e^z|<|z|$ when the real part of $z$ is negative

Prove that for any $z \in \mathbb{C}$, such that $\mathcal{R(z)}<0$, the inequality $|1-e^z|<|z|$ is true. I am thinking to use Maximal Modulus principle to prove this theorem. However, I am ...
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2answers
411 views

Definitions of analytic, regular, holomorphic, differentiable, conformal: what implies what and do any imply that a function is a bijection?

I'm looking back at some complex analysis and have gotten myself a little muddled in all of the definitions analytic/ regular/ holomorphic/ differentiable/ conformal... In particular, at the moment I'...
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218 views

What uniquely characterizes the germ of a smooth function?

Let $X$ be the set of all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ which are infinitely differentiable at $0$. Let us define an equivalence relation $\sim$ on $X$ by saying that $f\sim g$ if ...
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Extensions of real analytic functions with multiple variables

I wonder if my statements below are correct. Let $V$ be an open domain in $\mathbb{R}^d$, and $U$ an open domain in $\mathbb{C}^d$ with $V=\operatorname{Re}U:=\{\operatorname{Re}z:z\in U\}$. I want ...
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1answer
38 views

monotone real analytic function

Suppose the function $f:[a,b]\rightarrow \mathbb{R}$ is real analytic and monotonically increasing, i.e. for all $x,y\in [a,b]$ such that $x<y$ we have $f(x)\leq f(y)$. Further, suppose that $f$ is ...
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Find Where $f(z)=x^2+iy^2$ Differentiable And Analytic

Let $f(z)=x^2+iy^2$ find where it is differentiable and where it is analytic in $\mathbb{C}$ For a function to be differentiable at a point it should fulfil C-R equations, we have $u(x,y)=x^2$ and $...
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1answer
38 views

Local minimum of an analytic function

This is a follow-up to a previous question of mine. I know that any local minimum $x_0$ of a function $f : \mathbb{R}^n \rightarrow \mathbb{R}$ has positive semi-definite Hessian $H(x_0) \succeq 0$. ...
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1answer
43 views

Infinite sum of analytic functions becomes non-analytic

I am looking for a simple example that a series of analytic functions can become non-analytic. This is in the context of phase transitions, where one considers the analyticity of the partition ...
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33 views

Schottky's Theorem - Sup is Max and more

In an exercise I received there is the following question: Let $R>0$. Let $\mathcal{F}$ be the family of functions $f \colon \{z:|z|<R\} \to \mathbb{C}$ that are analytic avoid the values $0,...
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1answer
46 views

Functions of diagonal matrices

I am studying linear algebra from notes for a mathematical physics course and the section on diagonal matrices says that if $M$ is a diagonal matrix and $f$ is analytic, then $f(M)$ is equal to the ...
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3answers
80 views

Number of analytic functions such that $(f(\frac{1}{n}))^3=\frac{n}{n+1}.$

I want to find the number of analytic functions on the unit disk $\mathbb{E} = \{z:|z|<1\}$ such that $f:\mathbb{E}\ \rightarrow \mathbb{C}$ is analytic and $(f(\frac{1}{n}))^3=\frac{n}{n+1}$ for ...
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1answer
36 views

Weaker hypothesis for the identity theorem for analytic functions

We know that, if $C\in\Bbb C$, $f$ is analytic in a domain $\Omega$ and there exists $\Lambda\subset \Omega$ with an accumulation point in $\Omega$ such that $f(z)=C$ for all $z\in \Lambda$ then $f\...