Thursday, 28 August 2014

integration - Evaluating intinfty0fraccosaxcoshxdx



0cosaxcoshxdx



a is a real number.
Hint: Consider a suitable rectangular contour




I know that 1coshz has simple poles when z=(2n1)π2i.



What should I do next?
I would appreciate any help. Thank you.


Answer



First off, let us notice that the function cos(ax)/cosh(x) does not change under xx. So,



0cosaxcoshxdx=12cosaxcoshxdx.



Next, we should take the contour γM to be like that. Line (1) is a part of real line [M,M], and the upper line (3) is the same line translated by πi.



The integral over line (3) could be rewritten as



(3)cosaxcoshxdx=(1)cosa(x+πi)cosh(x+πi)dx=().




Notice, that cosh(x+πi)=coshx and



cosa(x+πi)=cos(ax)cos(aπi)sin(ax)sin(aπi).



Therefore,



()=cos(aπi)(1)cosaxcoshxdx+sin(aπi)(1)sinaxcoshxdx.



Last integral clearly equals zero, as it is the sum of symmetric and antisymmetric function over symmetric interval.



Integrals over lines (2) and (4) go to zero as M. So, we have now, via residuals theorem



2πirescosaxcoshx=lim


No comments:

Post a Comment

real analysis - How to find lim_{hrightarrow 0}frac{sin(ha)}{h}

How to find \lim_{h\rightarrow 0}\frac{\sin(ha)}{h} without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...