Questions tagged [analytic-functions]

Question 1

DEFINITION Let $ X$ and $ E$ be Banach spaces. A map $f \colon X\to E $ is analytic at $ x=0$ if $$ f(x)=\sum_{k=0}^{\infty}a_k(x) $$ where for each $ k$, the map $a_k \colon E^k\to E $ is ...

Question 2

I know when finding the order of a zero, $z_0$, of an analytic function, you can just differentiate the function until substituting $z_0$ in does not give zero. Instead, to save time on ...

Question 3

Let $a_1,a_2,\ldots$ be an infinite number of isolated poins in $\mathbb{C}$. Let $U=\mathbb{C} \setminus \{a_1,a_2,\ldots\}$ and $f:U \to \mathbb{C}$ be holomorphic. Assume that $\lim_{z \to a_k} (z-...

Question 4

In the Pólya vector field representation of a complex function f(z), the field is defined using the complex conjugate of , i.e. conjugate(f(z)) At first glance, this seems counterintuitive — wouldn’t ...

Question 5

Here is the definition of an analytic map. ANALYTIC MAP Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if $$ f(x)=\sum_{k=0}^{\infty}a_k(x) \tag 1 $$ where for ...

Question 6

Here are two definitions. ANALYTIC MAP Let $ E$ and $ F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $ x=0$ if $$ f(x)=\sum_{k=0}^{\infty}a_k(x) $$ where for each $ k$, the map $a_k \...

Question 7

The following is one version of Morera's theorem from complex analysis, as presented by Theodore W. Gamelin. Theorem (Morera’s Theorem). Let $f(z)$ be a continuous function on a domain $D$ (defined as ...

Question 8

I encountered the following theorem in one of my old complex analysis classes: (1) Suppose that $F(z) = \sum_{n = 1}^{\infty} a_n z^n$ is an analytic function with $a_0 = 0$. Then, there exists $R' &...

Question 9

Let $f:\mathbb R\to\mathbb R$ real analytic function on $\mathbb R$. Suppose that its Taylor series around $x=0$ is $$ S(x) = \sum_{k=0}^\infty (-1)^k\,a_k\,x^k$$ where $a_k\geq0$ for all $k$. In ...

Question 10

Let $f:\mathbb R\to\mathbb R$ real analytic function. Suppose that $f^{(k)}(0)$ has sign $(-1)^{k}$ for every $k=0,1,2,\dots$ Suppose also $\lim_{x\to\infty}f(x)=0$. Can we say that $f(x)\geq0$ for ...

Question 11

I have a double integral, which I want to solve: $\mathcal{I}_{au}=\int_0^{t'}e^{(a-c)t''}\int_0^{t''}e^{(-b+c)t'''}\text{exp}\big(-\frac{\lambda^2}{2}\left( S_2(t''')^2+S_1(t')^2+S_1(t'')^2+2S_{12}t'...

Question 12

Let $A(s)$ be an $N\times N$ matrix with all its elements proper rational functions in $s$ with real coefficients, and are analytic in the closed right-half plane (RHP) $\mathrm{Re}(s)\geq0$, i.e., ...

Question 13

Analytic map Let $E$ and $F$ be Banach spaces. A map $f \colon E\to F$ is analytic at $0$ if it can be written as $$ f(x)=\sum_{n=0}^{\infty}\alpha_n(x) $$ where $\alpha_n$ is a continuous $n$-...

Question 14

Definition Let $E,F$ be Banach spaces. A map $f \colon E\to F $ is analytic at $x_0\in E$ if it can be written as $$ f(x)=\sum_{n=0}^{\infty}\alpha_n(x-x_0) $$ where $\alpha_n$ is a symmetric ...

Question 15

If a function $f$ can be expressed using the Taylor expansion: $$ f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!} x^k $$ it must necessarily be analytic. For $f$ to satisfy a Hermite expansion, $f$ ...

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Questions tagged [analytic-functions]

analytic map on the Banach $X\times Y$

Finding zeros of the product of two analytic functions

Extension of an holomorphic function with an infinite number of removable singularities

Why does the Pólya vector field use the conjugate of a complex function?

Unconditional convergence and analytic map

Associativity of the sum in an analytic function

Can Analyticity Extend to the Boundary in Morera’s Theorem?

Criteria for congruent power series

A condition on all derivatives at $x=0$ implies $f\geq0$?

All derivatives with alternating signs at $x=0$ imply $f\geq0$?

I need help with solving a double integral exp of a quadratic variable

Existence of real eigenvalue > 1 on the imaginary axis for a rational matrix with a RHP eigenvalue of 1

differential of analytic map between Banach spaces

Normal convergence of the differentials for an analytic function

Why is the Hermite expansion more powerful than the Taylor expansion?

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