Timeline for Integration of $\int\limits_0^{2\pi} \int\limits_0^{2\pi} \min \{ x-y, 2\pi- (x-y) \} e^{a\cos(x)} e^{a\cos(y)} \, dx \, dy$

10 events

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S May 7 at 16:21	history	suggested	CommunityBot	CC BY-SA 4.0	separating du and dv
May 7 at 13:22	comment	added	Michael Hardy		Next: $$ \begin{align} & \int_{-v}^v \min\{v\sqrt{2}, 2\pi-v\sqrt{2}\} e^{a\cos\left(\frac{u+v}{\sqrt{2}}\right)} e^{a\cos\left(\frac{u-v}{\sqrt{2}}\right)} \, du \\ {} \\ = {} & \min\{v\sqrt{2}, 2\pi-v\sqrt{2}\} \int_{-v}^v e^{a\cos\left(\frac{u+v}{\sqrt{2}}\right)} e^{a\cos\left(\frac{u-v}{\sqrt{2}}\right)} \, du \end{align}$$ Then think about a trigonometric identity whose left side is the sum of two cosines.
May 7 at 13:15	review	Suggested edits
S May 7 at 16:21
Dec 18, 2021 at 20:33	comment	added	Henri Cohen		The (simple) integral of $\exp(a\cos(x))$ is equal to $2\pi I_0(x)$ I believe. Is there an "explicit" formula for the integral of $x\exp(a\cos(x))$ ? This would be a first step.
Dec 18, 2021 at 20:32	review	Low quality posts
Dec 19, 2021 at 11:25
Dec 18, 2021 at 20:29	comment	added	Alex M.		I am afraid the your result isn't any easier to work with than the original integral, in particular because now you have a combination of $u$ and $v$ inside the trigonometric functions, which in turn are inside the exponential.
S Dec 18, 2021 at 20:16	review	First answers
Dec 18, 2021 at 20:30
S Dec 18, 2021 at 20:16	history	edited	ors	CC BY-SA 4.0	added 12 characters in body
S Dec 18, 2021 at 20:08	review	First answers
Dec 18, 2021 at 20:10
S Dec 18, 2021 at 20:08	history	answered	ors	CC BY-SA 4.0