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Exp-trig integral and Lambert W-function

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Post Mon Feb 16, 2015 5:41 am

Posts: 50
Location: Redmond, WA

Prove:
$${\large\int}_0^\pi\frac{\cos4x-\cos2x-2\,e^{\cos2x}\,\sin(5x-\sin2x)\,\sin x}{1+e^{2\cos2x}+2\,e^{\cos2x}\cos(2x-\sin2x)}\,dx=\pi\,W(1),$$
where $W(z)$ is the Lambert W-function.


Posts: 50
I actually obtained a different (but similar) integral with the same result.

Observe that
$$\oint_{|z|=1}\frac{z(e^z+1)}{e^z+z}dz=2\pi i\operatorname*{Res}_{z=-W(1)}\frac{z(e^z+1)}{e^z+z}=-2\pi iW(1)$$
Parameterising the contour integral gives
\begin{align}
\Im\oint_{|z|=1}\frac{z(e^z+1)}{e^z+z}dz
=&\ \Re\int^{2\pi}_0\frac{(\cos(2x)+i\sin(2x))(1+e^{\cos{x}}\cos(\sin{x})+ie^{\cos{x}}\sin(\sin{x}))}{e^{\cos{x}}\cos(\sin{x})+\cos{x}+i(e^{\cos{x}}\sin(\sin{x})+\sin{x})}dx\\
=&\ \Re\int^{2\pi}_0\frac{\cos(2x)+e^{\cos{x}}\cos(2x+\sin{x})+i(\sin(2x)+e^{\cos{x}}\sin(2x+\sin{x}))}{e^{\cos{x}}\cos(\sin{x})+\cos{x}+i(e^{\cos{x}}\sin(\sin{x})+\sin{x})}dx\\
=&\ \int^{2\pi}_0\frac{e^{\cos{x}}\cos(2x-\sin{x})+\cos{x}+e^{2\cos{x}}\cos(2x)+e^{\cos{x}}\cos(x+\sin{x})}{e^{2\cos{x}}+2e^{\cos{x}}\cos(x-\sin{x})+1}dx
\end{align}
and $x\mapsto 2x$ yields
$$-\int^{\pi}_0\frac{\cos(2x)+e^{2\cos(2x)}\cos(4x)+2e^{\cos(2x)}\cos(3x)\cos(x-\sin(2x))}{e^{2\cos(2x)}+2e^{\cos(2x)}\cos(2x-\sin(2x))+1}dx
=\pi W(1)$$
It remains to "guess" the right function to integrate.


Posts: 1
Well done if I may!

Let Wo be the upper branch of the Lambert fct. of a real argument...
Can someone help me compute:

integral( Wo( a exp(b cos(u)) ) ) from u=0 to u=pi, please?


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