# Tagged Questions

Questions about the properties of functions of the form $\sum_{n=0}^{\infty}a_n x^n$, where the $a_n$ are real or complex numbers, and $x$ is real or complex (or more generally an element of a Banach algebra).

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### Determining uniform convergence of complex power series

I'm working on some practice problems for my complex analysis course, and I'm having trouble with uniform convergence. The question asks whether the following series converges uniformly for $|z|<1$:...
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### Power series with simple recurrence relationship: $a_{n+2} = a_{n+1} - \frac{1}{4}a_n$. How to determine corresponding closed form function?

Given: $$\sum_{n = 0}^{\infty} a_nx^n = f(x)$$ where: $$a_{n+2} = a_{n+1} - \frac{1}{4}a_n$$ is the recurrence relationship for $a_2$ and above ($a_0$ and $a_1$ are also given). Is there a nice ...
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### Borel-/Laplace-transform and $\psi$-function

I'm considering some family of functions whose coefficients of their power series occur in the columns of the following matrix A (of course thought as of infinite size) $\qquad$ The ...
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### Multiplication of a complex function with essential singularity with another complex function with a pole at the same point

Im trying to proof or disprove the following claim: If $f(z)$ and $g(z)$ are holomorphic in an annulus $0 < |z − z(\beta)| < R$ and $f$ has an essential singularity at $z(\beta)$ and $g$ has a ...
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### Function such that $f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!}$

I was trying to solve another problem and come up with the problem if there is a function with closed form such that $$f^{(n)}(0)=\frac{(n!)^2}{(2n+1)!};(n\ge1).$$ I tried to check the condition for ...
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### Sum of the series $\sum_{i=1}^n a^i i^r$

How can I find the sum of the series : $$\sum_{i=1}^n a^ii^r$$
I was looking at the Partial Differential Equation involving function: $$z(x,y)$$ $$\frac{\partial z}{\partial x} + c \frac{\partial z}{\partial y} = 0$$ Which fairly intuitively has a solution:...
Consider $$f(x)=\sum_{n=0}^{\infty}\sum_{j=0}^{n}\sum_{k=0}^{j}d_{k}c_{j-k}a_{n-j}x^{n-11}+\sum_{n=0}^{\infty}\sum_{k=0}^{n}e_{k}b_{n-k}x^{n-8}.$$ Given that I know the first few values of $d,c,a,e,b$...