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#2435 No result for definite integral (IV)

open
nobody
5
2012-11-18
2012-07-01
No

Enter:

integrate(sin(a*x)/(sinh(%pi*x)),x,0,inf);

and Maxima returns:

integrate(sin(a*x)/sinh(%pi*x),x,0,inf)

The correct result is: (1/2)*tanh(a/2), a>0

build_info("5.27.0","2012-04-24 08:52:03","i686-pc-mingw32","GNU Common Lisp (GCL)","GCL 2.6.8")

Regards

Chris

Discussion

  • Aleksas
    Aleksas
    2012-07-10

    Example. integrate(sin(a*x)/sinh(pi*x),x,minf,inf)
    for details see
    http://www.seas.harvard.edu/ekaxiras/images/tamaroot.pdf

    (%i1) S:'integrate(sin(a*x)/sinh(%pi*x),x,minf,inf)$
    (%i2) assume(a>0)$
    (%i3) f(x):=exp(%i*a*x)/sinh(%pi*x);
    (%o3) f(x):=exp(%i*a*x)/sinh(%pi*x)
    (%i4) r:2*%pi*%i*residue(f(x),x,%i);
    (%o4) -2*%i*%e^(-a)
    (%i5) r1:-%pi*%i*residue(f(x),x,0);
    (%o5) -%i
    (%i6) r2:-%pi*%i*residue(f(x),x,2*%i);
    (%o6) -%i*%e^(-2*a)
    (%i7) 'integrate(f(x),x,minf,inf)+'integrate(f(x+2*%i),x,inf,minf)
    +r1+r2=r;
    (%o7) -integrate(%e^(%i*a*(x+2*%i))/sinh(%pi*(x+2*%i)),x,-inf,inf)+integrate(%e^(%i*a*x)/sinh(%pi*x),x,-inf,inf)-%i*%e^(-2*a)-%i=-2*%i*%e^(-a)
    (%i8) trigexpand(expand(%));
    (%o8) -integrate(%e^(%i*a*x-2*a)/sinh(%pi*x),x,-inf,inf)+integrate(%e^(%i*a*x)/sinh(%pi*x),x,-inf,inf)-%i*%e^(-2*a)-%i=-2*%i*%e^(-a)
    (%i9) imagpart(%);
    (%o9) -%e^(-2*a)*integrate(sin(a*x)/sinh(%pi*x),x,-inf,inf)+integrate(sin(a*x)/sinh(%pi*x),x,-inf,inf)-%e^(-2*a)-1=-2*%e^(-a)
    (%i10) solution:solve(%,S)[1];
    (%o10) integrate(sin(a*x)/sinh(%pi*x),x,-inf,inf)=(%e^a-1)/(%e^a+1)

    (%i11) 'integrate(sin(a*x)/sinh(%pi*x),x,0,inf)=(%e^a-1)/(%e^a+1)/2;
    (%o11) integrate(sin(a*x)/sinh(%pi*x),x,0,inf)=(%e^a-1)/(2*(%e^a+1))

    (%i12) sol1:rhs(%);
    (%o12) (%e^a-1)/(2*(%e^a+1))

    The correct result is: (1/2)*tanh(a/2)
    (%i13) sol2:(1/2)*tanh(a/2);
    (%o13) tanh(a/2)/2
    (%i14) exponentialize(%);
    (%o14) (%e^(a/2)-%e^(-a/2))/(2*(%e^(a/2)+%e^(-a/2)))
    (%i15) factor(%);
    (%o15) (%e^a-1)/(2*(%e^a+1))
    (%i16) is(%=sol1);
    (%o16) true
    Then sol1=sol2.

    Aleksas D

     
  • Thanks for your answer.

    A very sophisticated solution!

    What about the integral (see item #3538167, 2012-06-27 06:03, please)

    integrate(cos(x)/cosh(x),x,0,inf)= %pi/(2*cosh(%pi/2))?

    I’m quite sure that you will find a way…

    Kind regards

    Chris

     
  • Aleksas
    Aleksas
    2012-07-11

    example 6 integrate(cos(x)/cosh(x),x,minf,inf);
    see: http://answers.yahoo.com/question/index?qid=20091007162254AAZXzt5

    (%i1) f(x):=cos(x)/cosh(x);
    (%o1) f(x):=cos(x)/cosh(x)
    (%i2) S:'integrate(f(x),x,minf,inf);
    (%o2) integrate(cos(x)/cosh(x),x,-inf,inf)
    (%i3) r:2*%pi*%i*residue(f(x),x,%pi/2*%i);
    (%o3) 2*%pi*cosh(%pi/2)
    (%i4) 'integrate(f(x),x,minf,inf)+'integrate(f(x+%pi*%i),x,inf,minf)=r;
    (%o4) integrate(cos(x)/cosh(x),x,-inf,inf)-integrate(cos(x+%i*%pi)/cosh(x+%i*%pi),x,-inf,inf)=2*%pi*cosh(%pi/2)
    (%i5) realpart(%);
    (%o5) cosh(%pi)*integrate(cos(x)/cosh(x),x,-inf,inf)+integrate(cos(x)/cosh(x),x,-inf,inf)=2*%pi*cosh(%pi/2)
    (%i6) ats:solve(%,S)[1];
    (%o6) integrate(cos(x)/cosh(x),x,-inf,inf)=(2*%pi*cosh(%pi/2))/(cosh(%pi)+1)
    (%i7) float(%), numer;
    (%o7) integrate(cos(x)/cosh(x),x,-inf,inf)=1.252040331252148
    (%i8) 2*quad_qagi(cos(x)/cosh(x), x, 0, inf);
    (%o8) [1.252040331252148,5.5816090477577717*10^-9,450,0]
    (%i9) sol1:rhs(ats);
    (%o9) (2*%pi*cosh(%pi/2))/(cosh(%pi)+1)
    (%i10) sol2:%pi / cosh(%pi/2);
    (%o10) %pi/cosh(%pi/2)
    (%i11) float(%), numer;
    (%o11) 1.252040331252148
    (%i12) trigrat(sol1-sol2);
    (%o12) 0

    best

    Aleksas D