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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Finding zeros of the product of two analytic functions
I know when finding the order of a zero, $z_0,ドル 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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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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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,ドル the map $a_k \...
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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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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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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,ドル i.e., ...
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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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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 ...