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Improper Integral $\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx$

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#integration #definite-integrals #improper-integrals #special-functions #closed-form
607
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Integral of combination of power, exponential, and confluent hypergeometric function

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#integration #sequences-and-series #power-series #special-functions #hypergeometric-function
567
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The derivatives of Riemann xi function

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#analysis #special-functions #taylor-expansion
635
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Show that $J_n(x)$ satisfies Bessel equation $ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $

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#real-analysis #special-functions
28
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Identifying a function

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#special-functions
411
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How can I show that this Jacobi polynomial can be expressed as the sum of these two Legendre polynomials?

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#special-functions #orthogonal-polynomials
60
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Numerical integration of $E_1(x)$

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#special-functions #indefinite-integrals
227
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Integral involving power of trigonometric functions

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#integration #special-functions #residue-calculus #hypergeometric-function #bessel-functions
184
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Refences to Sturm–Liouville theory with a singular weight function.

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#special-functions #orthogonality #orthogonal-polynomials #sturm-liouville
42
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Convergence of series

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#sequences-and-series #convergence-divergence #power-series #special-functions #uniform-convergence
113
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Simple form of LegendreQ function

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#special-functions
71
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How to prove this relation?

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#limits #special-functions #legendre-polynomials
69
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Can you justify the existence of a $x_{*}$ solving $\mbox{li}(x_{*})=\mbox{erf}(x_{*})?$

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#analysis #numerical-methods #special-functions
286
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May I know how this integral was evaluated using hypergeometric function?

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#integration #special-functions #hypergeometric-function
4.6k
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When actually $f(g(x))=g(f(x))$ holds?

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#functions #special-functions

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