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6h
answered Closed form of a series (dilogarithm)
8h
comment How to prove that $J_\frac{-5}{2}(x)= \frac{\sqrt2}{\sqrt{x\pi}}[\frac{3}{x}\sin x+\frac{3-x^2}{x^2}\cos x]$
In general, $x^a\cdot\sqrt{\dfrac\pi2}\cdot J_{-a}\big(x\big)~=~P_k(x)~\sin x+Q_k(x)~\cos x$, where $a=k+\dfrac12.$
12h
comment Integrals with error function and exponentials
First of all, you should differentiate with regard to the parameter m, to get rid of the $\dfrac1{\sqrt{x^2+y^2}}$ in the integrand. Secondly, the very presence of $\sqrt{x^2+y^2}$ in the exponent begs for a trigonometric or hyperbolic substitution. Unfortunately, the only functions which are defined in this way are the Bessel and Struve functions.
22h
comment Evaluating $\int_0^{\pi /2}\left(\frac{1}{\sqrt{\tan(x)}}+\frac{1}{\sqrt{\arctan(x)}}\right) dx$
See Wallis' integrals.
1d
comment How to approximate $x^y$ using a quadratic function
For $x\in\bigg[\dfrac23,\dfrac32\bigg]$, we have $\ln x\simeq\sqrt x-\dfrac1{\sqrt x}$ .
1d
answered Evaluating $\int\sqrt{\frac{1-x^2}{1+x^2}}\mathrm dx$
1d
comment Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide
This is conceptually similar to a Mobius transformation.
1d
comment Closed-form of $\operatorname{Li}_2\left(1 \pm i\sqrt{3}\right)$
the essential idea is exactly the same - Though their equivalence may seem crystal clear to you, it certainly does not appear so to others, $($such as myself, for instance$)$, which is why your approach is profitable to many readers.
1d
comment Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator
Can you also prove that $\gamma<\dfrac1{\sqrt3}$ ? :-$)$
1d
comment Diophantine equation: $13^x+3=y^2$
$x=0$ and $x=1$ are the only solutions.
1d
comment Find the maximum volume of the cylinder.
Hint: Given what you already wrote, it is clear that the cylinder of maximum volume corresponds to the rectangle of maximum area.
1d
answered Equation to get the center point of the union of n ellipses?
1d
comment Hankel transform of a Bessel function of different order
A related question.
1d
answered definite integral of $x^2e^{-x^2}$
1d
comment $AB=AC$, $BD$ is the angle bisector of $\angle B$ , find $\angle A$
For humorous effect, replace “Stewart's theorem” with “Jupiter”. :-$)$
1d
comment solve nonlinear system of equation numerically
Copy-paste the $\LaTeX$ code into Desmos.
2d
comment Finding the Derivative of $\sqrt{x}$
Hint: $x-x_0=\big(\sqrt x\big)^2-\big(\sqrt x_0\big)^2=\big(\sqrt x-\sqrt x_0\big)~\big(\sqrt x+\sqrt x_0\big)$
2d
comment Evaluating the ratio $ {{a_{n+1}}\over{a_n}}$ in calculating the radius of convergence for a power series
As an aside, $\displaystyle\sum_{n=0}^\infty{2n\choose n}~x^n~=~\frac1{\sqrt{1-4x}}$ . See binomial series for more information.
2d
comment Binomial Sum Formula
This is Vandermonde's identity.
Aug
25
revised Intuition behind $(-\frac{1}{2})! = \sqrt{\pi}$
Row Alignment.