# 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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### Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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### Laurent-Series on an annulus

I solved an exercise and I want to know if it is correct. I'm trying to find the Laurent-Series for $$g(w)=\frac{w}{1+w^2}$$ On the annulus $D_{1,2}(-i)$ What I did so far: We have 2 poles in $w=i$ ...
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### Singular expansion of an implicit function

In the book of Flajolet and Sedgewick (this context is not so important, though), the following argumentation is used: Let $y(z)$ be a function given implicitly by $y - \phi(z,y) = 0$, where $\phi$ ...
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### $\sum _{j=0}^{\infty }\binom{-p-1}{j} \bigl( -\frac {x} {1+x}\bigr) ^{j}=?$

I did try to use geometric series somehow. I have no idea how to evaluate in terms of $p$ and $x$.
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### Deriving the additive property of natural log from power series?

I know the additive property of logarithms, that $$\ln(x) + \ln(y) = \ln(xy)$$ is easy to prove using the logarithm's nature as the inverse of the exponential function. However, I'm interested in ...
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### Radius of convergence and the existence of antiderivative

I think I have some misunderstandings regarding some basic concepts. First, the question I'm dealing with is the following: Let $f$ be analytic in $\{z ;|z|>1 \}$, and $\int_{|z|=2}f(z)dz=0$. ...
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### Series expansion of $1/(1+z^2)$ about the point I

I am trying to find a series representation for the complex function: $1/(1+z^2)$. The text I am reading gives: $1/(1+z^2) = 1/((z+I)(z-I)) = -I/(2(z-I)) +1/4 - I(z-I)/8 - (z-I)^2/16 + ...$ I do not ...
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### Multidimensional taylor series $sin (x^3y^2)$

A homework of mine was to compute the Taylor series of $f(x,y)=\sin(x^3y^2)$ around $(0,0)$ to the 25th order. I assumed, as $\sin(z)=\sum\limits^{\infty}_{k=0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}$, that I ...
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### Maclaurin Expansion of $\ln(3+x)$

I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: $\ln\left[3\left(1+\dfrac{x}{3}\right)\right]$ as opposed ...
### What is the power series expansion at $x=0$ of the algebraic function defined by $(27x-4)y^3 + 3y + 1 = 0$?
Let $y$ denote the complex-valued algebraic function defined implicitly near $x=0$ by $(27x - 4)y^3 + 3y + 1=0$ and such that $y(0)=1$. What is the power series expansion of this function at $x=0$? ...