All Questions
Tagged with complex-analysis power-series
1,470
questions
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The power series $\sqrt{1-z}$ about zero converges absolutely for all $|z| \leq 1$
I am following a proof for this claim (from Reed & Simon's book on functional analysis) which lets
$$\sqrt{1-z} = 1 + c_1 z + c_2 z^2 + \ldots$$
be the power series of $\sqrt{1-z}$ about the ...
- 6,277
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Multiplication of multiple summations of complex functions
I have a series that looks like $\sum_{l,m,n}\frac{A^{l}B^{m}C^{n}}{l!m!n!}$ where $A$ is a complex function and $B$ and $C$ are real functions. The summation is finite up to some cutoff $p$. $A$, $B$,...
- 21
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2
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Coefficients of a product of a Laurent polynomial and an infinite series
Let $N$ be an integer. Let us consider a Laurent polynomial in $q$ given by $\sum_{s=N}^{M} \gamma_s q^s$. Then consider the expression $(\sum_{s=N}^{M} \gamma_s q^s) \times \sum_{i=0}^{\infty}q^i$.
...
- 443
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(Complex) Power series: two different but equivalent definitions for radius of convergence? [duplicate]
In complex analysis the definition of radius of convergence is as far as I know given by
$$
R = \mathrm{sup} \left\{ |z| : \sum_{n=0}^{\infty} |a_n z^n| \;\text{converges} \right\}
$$
So this ...
- 331
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Polynomial approximation on the closed unit disc
Let $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ and let $A$ be the Banach algebra of functions that are analytic in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$ with the uniform norm. I want to ...
- 1,926
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2
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73
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Solving Power Series Equations if we introduce Logarithmic Terms
If we have a complex power series equation like
$$
\sum_{n=0}^{\infty} a_n z^n = \sum_{n=0}^{\infty} b_n z^n
$$
then we can conclude $a_n = b_n$. We can see this by viewing $z^n$ as basis elements, or ...
- 351
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Why this equation holds?
Can someone help me why does this equation holds:
$\sum_{\rho}(1/2 + t -\rho)^{-s}= e^{i\pi s/2}\sum_{k=1}^{\infty}(\tau_k+it)^{-s}+e^{-i\pi s/2}\sum_{k=1}^{\infty}(\tau_k-it)^{-s}$ for $\rho=1/2 \pm ...
- 87
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Combinatorics: Singular Inversion Theorem vs Smooth Implicit-Function Schema
In lecture I saw the following theorem:
Singular Inversion Theorem: Let $\phi(u)$ be analytic at $0$ and its series expansion $\phi(u) = \sum_{k} \phi_k u^k$ at $0$ satisfy the following conditions:
...
- 4,222
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Is this the expansion for any known function?
The expansion is $$\sum_{n=1}^{\infty}\frac{x^n}{n!(n-1)!}\left[c(1+c)\dots((n-1)^2+c)\right]$$
So the first 3 terms are $cx$, $\dfrac{c(1+c)}{2}x^2$, $\dfrac{c(1+c)(4+c)}{12}x^3$.
- 35
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votes
1
answer
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Can this given $f: S^1\to \mathbb C$ be extended to a continuous $F: \overline{\mathbb D}\to \mathbb C, F$ is holomorphic on $\mathbb D$?
Suppose that $f: \mathbb S^1\to \mathbb C$ is continuous such that $f(z)=f(\bar z)$ for all $z\in \mathbb S^1$. Can it be extended to a continuous $F: \overline{\mathbb D}\to \mathbb C$ such that $ F$ ...
- 11.5k
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1
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105
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Is univalent polynomials dense in $\mathbb S$?
Let $\mathbb S$ be the collection of all univalent and analytic functions defined on unit disc $\mathbb D$ such that for each $f \in \mathbb S,$ we have $f(0)=0, f'(0)=1.$
Let $P$ be the subsets of $\...
- 751
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64
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Finding the Maclaurin series for the given function.
I am dealing with a problem that says:Find the Maclaurin series for the function $f(x)=\ln(1+x+x^2)$.
I have tried to write the expression inside the brackets exactly, $1+x+x^2$ as a product of two ...
- 5
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1
answer
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$g_2(z)=g_1(z)+O(z^n)$ implies $1/g_2(z)=1/g_1(z)+O(z^n)$
Let $f_2,g_2:B_r(0) \longrightarrow \mathbb{C}$ be two holomorphic functions such that $f_2(z)=f_1(z)+O(z^n)$ and $g_2(z)=g_1(z)+O(z^n)$, here $O(z^n)$ denotes a power series of the form $\sum_{k=n} ...
- 630
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if $P(x_{0})=0$, but $\lim_{ x \to x_{0} }(x-x_{0})\left( \frac{Q(x)}{P(x)} \right) = \text{finite}$, then function is analytic at $x=x_0$
I'm currently learning about Differential Equations and the textbook I'm using (by DiPrima) says the following:
For polynomials $P(x)$ and $Q(x)$ that has no common factors, if $P(x_{0})=0$, but $\...
- 1,688
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Hadamard Gap Theorem and Lacunary Functions
Does anyone know any good reference or even a simple proof for Hadamard Gap Theorem or even just the fact that a lacunary function diverges at 1 (I mean the limit not just the evaluation). In fact, I ...
- 1,179