All Questions
Tagged with complex-analysis power-series
1,470
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Disk of convergence of 3 power series. [duplicate]
I am given 3 power series and I want to find (if possible) the region of the complex plane in which the power series converge at the same time. The power series are
$\displaystyle \sum_{n=1}^{\infty} \...
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Functions in disk algebra with summable series coefficients
We know that there are functions $f \in {A}(\mathbb{D}) = C(\overline{\mathbb{D}}) \cap \mathrm{Hol}(\mathbb{D})$ such that $f(z) = \sum_{k\geq0}a_k z^k$ on $\mathbb{D}$ with $a=(a_k)_{k\geq0} \notin \...
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Radius of Convergence of $\sum_{n=1}^{\infty}\frac{z^{3^n}-z^{2\cdot 3^n}}n$
Consider the power series
$$g(z)=\sum_{n=1}^{\infty} \frac{z^{3^n}-z^{2 \cdot 3^n}}{n}.$$
We can see that $\limsup_{n \to \infty} |a_n|^{1/n}=0$ so by Cauchy-Hadamard we know that the radius of ...
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How to prove $\sum\nolimits_{k = 0}^\infty {{b_k}/{{(z - {z_0})}^k}} $ is uniformly convergence for all z in that annulus
I read this in my book
By substituting the reciprocal of $\left( {z{\rm{ }} - {\rm{ }}{z_0}} \right)$in the power series, we can show
that if $\sum\nolimits_{k = 0}^\infty {{b_k}/{{(z - {z_0})}^k}} $...
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1
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If $f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ has radius of convergence $1$ and all $a_n\geq 0$, then $z=1$ is a singular point of $f$.
Suppose $f(z)=\sum\limits_{n=0}^{\infty}a_nz^n$ has convergence radius $1$. If all $a_n\geq 0$, then $z=1$ is a singular point of $f$.
The only proof I have seen can be found here. I have been trying ...
- 1,777
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Holomorphic approximation of a non-holomorphic function
I am looking at "near-holomorphic" continuous bounded functions defined from an open disc $D$ to $\mathbb{C}$ such that the Cauchy Riemann equations aren't exactly satisfied but the ...
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2
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Expanding a power series at a different point
Let $f(z)=\sum_na_nz^n$ be a complex power series with convergence radius $R$, I want to show the function $f$ is analytic, i.e. can be expanded to a power series on a neighbourhood of any point in ...
- 1,434
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Complex Analysis Qualifying Exam Problem Regarding Taylor Series and Normal Convergence
I am studying for a qualifying exam on Gamelin's Complex Analysis Chapters 1-11 and am stuck on the following past exam question:
Let $\phi(n): \mathbb{N} \to \mathbb{R}$ such that $\lim_{n\to \infty} ...
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Radius of convergence for a power series where coefficients are given by recursive relation
Problem: Let $f(z) = \sum_{n \geq 0} a_n z^n$ be the formal power series with coefficients given by the following recursive relation: given $\alpha, \beta \in \mathbb{R}$,
$$a_0 = 0$$
$$a_1 = 1$$
and
$...
- 312
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1
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99
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can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$?
can we have a Taylor series expansion of the function $\frac{1}{\zeta(s)}$at $s=1$?
if so how can we determine the radius of convergence of this expansion without assuming the truth of the riemann ...
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2
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Boundary limit of a typical holomorphic function with natural boundary being the unit circle
The power series $f(z)=\sum_{n\ge 1}z^{n!}$ defines a holomorphic function on the unit disk $D=\{z\in \mathbb{C}:|z|<1\}$ in the complex plane $\mathbb{C}$. It has the unit circle as the natural ...
- 1,308
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Jameson complex analysis Exercise 1.2.5: $\sum_{n=0}^{\infty} a_n z^n = s(z)$ is real for all real $z$, prove $a_n$ are real
How to solve the following problem? I can't use differentiation to solve it because this exercise is presented before the section that introduces differentiation in the book.
Suppose that $\sum_{n=0}^{...
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Relationship between ${}_3F_2$ hypergeometric functions
The ${}_3F_2$ hypergeometric function is defined as:
$$ {}_3F_2(1,a,b;c,d;z)=\dfrac{\Gamma(c)\Gamma(d)}{\Gamma(a)\Gamma(b)}\sum_{n=0}^{\infty}\dfrac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)}z^n,$...
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How many terms can you remove from the power series $f(x)$ and still have an extendable function?
I was watching this video and I had a question. The video explains how
$f(z) = 1 + z + z^2 + \cdots$
has an analytic continuation (that being $\frac{1}{1-z}$) but the function with just the powers of ...
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2
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Prove that the complex numbers for which the series $\sum_{k=1}^{\infty} \frac{3^n}{z^n+z^{-n}}$ converges is an open set of $\mathbb{C}$
My ideas was to find the radius of convergence of the series using the ratio test that says
$$R = \frac{1}{\lim_{n\to\infty}|\frac{a_{n+1}}{a_n}|} = \lim_{n\to\infty}|\frac{a_n}{a_{n+1}}|$$
However, I ...
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