# Verifying a cosine FourierTransform

On page 31 of this standard reference we have the following relation:

I wanted to use mathematica to verify this transform numerically for some example values. So I type in:

transf[a_, c_, \[Beta]_, w_] =
(\[Pi] (Cos[\[Pi]/\[Beta] -
( a c)/\[Beta]] Cosh[\[Pi]/\[Beta] + (c w)/\[Beta]] -
Cos[\[Pi]/\[Beta] + (a c)/\[Beta]] Cosh[\[Pi]/\[Beta] -
( c w)/\[Beta]]))/(\[Beta] Sin[ c] (Cosh[(2 \[Pi] w)/\[Beta]] -
Cos[(2 \[Pi] a)/\[Beta]]))


And try, i.e. the following

NIntegrate[Cosh[a x]/(Cosh[\[Beta] x]+Cos[c])Cos[x Zeta[3]]
/.a->0 /.\[Beta]->2/.c->\[Pi]/3,{x,0,Infinity}]


0.377469

Trying to evaluate the analytic right hand side at the same point gives however:

transf[0, \[Pi]/3, 2, Zeta[3]]


0

Which is clearly not the same. What is the issue here? Is the numeric integration in Mathematica wrong? Is there a typo in the reference? Did I misunderstand some of the input? Thanks for any suggestion!

• FWIW FourierCosTransform produces an analytic result for c=Pi/2 , a=0 that agrees with your NIntegrate and not with the reference expression. – george2079 Apr 17 '17 at 20:10