Questions tagged [analytic-functions]
For questions about analytic functions, which are real or complex functions locally given by a convergent power series.
1,377 questions
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Finding zeros of the product of two analytic functions
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 ...
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Extension of an holomorphic function with an infinite number of removable singularities
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-...
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Why does the Pólya vector field use the conjugate of a complex function?
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 ...
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Unconditional convergence and analytic map
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 ...
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Associativity of the sum in an analytic function
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 \...
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answer
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Can Analyticity Extend to the Boundary in Morera’s Theorem?
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 ...
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Criteria for congruent power series
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' &...
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A condition on all derivatives at $x=0$ implies $f\geq0$?
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 ...
- 951
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All derivatives with alternating signs at $x=0$ imply $f\geq0$?
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 ...
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I need help with solving a double integral exp of a quadratic variable
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'...
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Existence of real eigenvalue > 1 on the imaginary axis for a rational matrix with a RHP eigenvalue of 1
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., ...
- 565
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differential of analytic map between Banach spaces
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 $n$-...
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Normal convergence of the differentials for an analytic function
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 ...
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Why is the Hermite expansion more powerful than the Taylor expansion?
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, $f$ ...
- 3,177
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Analytic continuation and the Poincaré-Volterra theorem
I'm reading the analytic continuation section of Shabat's book. There is a theorem he calls "Poincaré-Volterra", which roughly says that the number of sheets of an analytic function is at ...
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