Tuesday, 29 September 2015

integration - Does intinfty0cos(coshx)cosh(alphax),mathrmdx converge for $0 le alpha



I highly suspect that 0cos(coshx)cosh(αx)dx

converges for 0α<1



(If true, it obviously also converges for 1<a<0.)




I can show that the integral converges for α=0:
0cos(coshx)dx=1cos(u)u21du


which converges by Dirichlet's test



I can also show that the integral doesn't converge for α=1:



0cos(coshx)cosh(x)dx=1ucos(u)u21du


which doesn't converge since ucos(u)u21cos(u) for large values of u



For other values of α between 0 and 1, I'm not sure what to do. I don't know how to express cosh(αx) in terms of cosh(x).



Answer



For α[0,1), let t=ex. Then



R0cos(coshx)cosh(αx)dx=eR1cos(t+t12)tα+tα2tdt.



Noticing that



cos(t+t12)=cos(t2)+O(12t)as t,




we find that the integral converges as R by Dirichlet's test.


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