All Questions
Tagged with complex-analysis power-series
311
questions with no upvoted or accepted answers
11
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0
answers
298
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Roots of partial sum of power series
Consider the power series
$$ \sqrt{1+z} = \sum_{k=0}^\infty \left( \begin{array}{c} \frac{1}{2} \\ k \end{array} \right) z^k. $$
I am interested in characterizing the roots of the partial sum
$$ s_n(z)...
- 1,436
11
votes
0
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253
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Find radius of convergence for a complicated series for $f'(f(x)) = f(f'(x))$
Several months ago, I answered this question asking for solutions to the functional equation $f'(f(x)) = f(f'(x))$ by expanding as a formal Taylor series around some arbitrary fixed point of $f$. This ...
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7
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131
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Geometry of the zeros of a power series.
This is probably a basic question that is easily googlable, but it seems that I dont have the right keywords. So my question is, having some power series
$$
f(z)=\sum_{k=0}^{\infty}C_{k}z^{k}, z\in\...
- 5,627
6
votes
0
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331
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Complex power series on the circle of convergence
I'm trying to find examples of complex power series $\sum a_n z^n$ with radius of convergence $1$ and where:
(1) The series converges everywhere on the circle $|z| = 1$ except one point;
(2) The ...
- 1,617
5
votes
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101
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Zeta Lerch function. Proof of functional equation.
so I'm trying to prove the functional equation of Lerch Zeta, through the Hankel contour and Residue theorem, did the following.
In the article "Note sur la function" by Mr. Mathias Lerch, a ...
5
votes
0
answers
221
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upper bound on $L_2$-norm of a power series, in terms of coefficients?
Is there any upper bound on $L_2$-norm of a convergent power series (in R), in terms of coefficients?
I have $f(x) = a_0+a_1\frac{x}{1!}+a_2\frac{x^2}{2!}+a_3\frac{x^3}{3!}+...$.
I need something like:...
- 341
5
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359
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Deriving a Series Representation of the Bessel Function of the First Kind
I've tried to use an integral representation of the Bessel Function of the First Kind $J_n(x)$ to derive a series representation of the function. My end result is pretty close to the answer that it ...
- 3,309
5
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answers
2k
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How is Lagrange's inversion theorem derived?
I am interested in the complex-analysis version of deriving Lagrange's inversion theorem:
If $y=f(x)$ with $f(a)=b$ and $f'(a)\neq 0$, then
$$x(y)=a+\sum_{n=1}^{\infty} \left(\lim_{x\to a}\...
- 59
4
votes
0
answers
113
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Finding radius of convergence on a Banach space
Let $(\phi_m)$ be the sequence of coordinate functionals on $\ell^p$ where $1\leq p <\infty.$ Then the power series $\sum_{m=0}^\infty (\phi_m(x))^m$ is absolutely convergent for any $x\in\ell^p$ ...
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4
votes
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402
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Conditions and correct interpretation of Borel summation
Hello to the community.
In my line of research (theoretical particle physics) it is customary to apply the strategy of Borel summation to infinite power series in order to find closed forms and/or ...
4
votes
0
answers
827
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Laurent expansion for $\sqrt{z(z-1)}$
Let $f(z) = \sqrt{z(z-1)}$. The branch cut is the real interval $[0,1]$, and $f(z)>0$ for real $z$ that are greater than 1. I need to find the first few terms of the Laurent expansion of $f(z)$ for ...
- 1,303
4
votes
0
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247
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Finding the radius of convergence of $\sum\limits_{j=0}^{\infty}2^jz^{j^2}$ using Cauchy-Hadamard formula: what am I doing wrong?
For a power series $\sum_{j=0}^{\infty}a_j(z-z_0)^j$ the Cauchy-Hadamard formula states that:$$R=\frac{1}{\operatorname{lim sup}\sqrt[n]{|a_n|}}$$
Where $\sqrt[n]{|a_n|}$ is a sequence formed from the ...
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4
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2k
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Does convergence of power series on radius of convergence imply absolute convergence?
Let $R$ be radius of convergence of power seires $\displaystyle\sum_{k}a_kz^k$.
If the power series converges for all $|z|=R$, can we say that it converges absolutely on the radius of convergence?
I ...
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4
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591
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Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?
Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
- 6,709
4
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Convergence of complex power series
I'll post the full problem before I'll show my (rather limited) progress:
i) Find all $z \in \mathbb{C}$ so that the following power series converge around $0$: a) $\sum_{k=0}^\infty z^k$, b) $\sum_{...
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