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Prove that $f(w)=\frac{1}{2\pi}\int_{\partial\Bbb{D}}\frac{f(z)}{1-w\bar{z}}|dz|$

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#complex-analysis #power-series
39
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Can the radius of convergence be equal to $1$?

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#power-series
209
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I have found a Taylor series and a Maclaurin series for a function about x=0.... I have a couple of questions.

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#sequences-and-series #convergence-divergence #summation #power-series #taylor-expansion
75
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How to show that $\frac{1}{w-z} = \frac{1}{w-a} + \frac{z-a}{(w-a)^2} + ... $, $w,z \in \mathbb{C}$

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#sequences-and-series #complex-analysis #power-series
52
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Finding the laurent series with an added imaginary component

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#sequences-and-series #power-series #laurent-series
55
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Two different series for $\frac{1}{x + 2}$

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#real-analysis #sequences-and-series #power-series
38
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Where does it come $\left[\sum_{n=0}^{\infty}x^n\right]' = \sum_{n=0}^{\infty}(n+1)x^n$

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#power-series
762
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Using termwise (term-by-term) differentiation on an infinite series to satisfy a differential equation.

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#sequences-and-series #ordinary-differential-equations #derivatives #power-series #taylor-expansion
43
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If $a,b,c$ are positive integers with $c\leq a+b,$ can I conclude that $_2F_1(a,b;c;1)$ diverges?

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#power-series #special-functions #gamma-function #hypergeometric-function
71
Views

Finding the sum $\sum_{n=0}^\infty\frac{(x+1)^{n+2}}{(n+2)!}$. I cannot use my usual methods that I am use to.

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#sequences-and-series #summation #power-series #taylor-expansion
1.1k
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Finding the first 3 terms of a power series... I am confused as to whether I can just plug numbers in or if there is more.

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#sequences-and-series #functional-analysis #summation #power-series #taylor-expansion
378
Views

Series expansion of $xy'=y$

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#ordinary-differential-equations #power-series #taylor-expansion
97
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Does the formula $\frac{1}{D + a} f = \frac{1}{a}\left(f - \frac{1}{D + a} f'\right)$ have a name?

Published on
#calculus #ordinary-differential-equations #power-series #differential-operators
68
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Application of power series expansion

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#calculus #complex-analysis #limits #convergence-divergence #power-series
85
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An ordinary differential equation equal to an infinite sum of legendre polynomials

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#sequences-and-series #ordinary-differential-equations #power-series #legendre-polynomials

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