Laplace Transform of Tanh[x] and Sech[x]^2

[str...@dirac.phys.washington.edu]

I ran across some peculiar behavior of Mathematica when doing the
Laplace Transform of Sech[x]^2 and Tanh[x].

Doing:

In[1]:= <<Calculus`LaplaceTransform`

In[2]:= Integrate[Tanh[x] * Exp[-s*x],{x,0,Infinity}]

gives the integral back again -> MMa can't do it but it can do

In[3]:= Integrate[Sech[x]^2 * Exp[-s*x],{x,0,Infinity}]

gives an answer that involves the PolyGamma[] function. Now
the derivative of Tanh[x] is Sech[x]^2 and

LT (d(tanh(x))/dx = s*LT tanh(x) - tanh(0) so I'd expect MMa to

do the LT of tanh(x) if it can do the LT of sech(x)^2.

I also checked with Maple. Maple cannot do the LT of tanh(x) and
sech(x)^2.

Nick Strobel
str...@dirac.phys.washington.edu

[Steven M. Christensen]

(* The appropriate use of LaplaceTransform is *)

In[1]:= <<Calculus`LaplaceTransform`

In[2]:= LaplaceTransform[Tanh[x],x,s]

1 s s
PolyGamma[0, - + -] - PolyGamma[0, -]
1 2 4 4
Out[2]= -(-) + -------------------------------------
s 2

In[3]:= Integrate[Sech[x]^2 * Exp[-s*x],{x,0,Infinity}]

s 2 + s
2 - s PolyGamma[0, 1 + -] + s PolyGamma[0, -----]
4 4
Out[3]= -------------------------------------------------
2

In[4]:= Expand[%3 - s * %2]

1 s s
s PolyGamma[0, - + -] s PolyGamma[0, 1 + -]
2 4 4
Out[4]= 2 - --------------------- - --------------------- +
2 2

s 2 + s
s PolyGamma[0, -] s PolyGamma[0, -----]
4 4
> ----------------- + ---------------------
2 2

In[5]:= Simplify[%]

s s
4 - s PolyGamma[0, 1 + -] + s PolyGamma[0, -]
4 4
Out[5]= ---------------------------------------------
2
(* Use the standared digamma identity *)

In[6]:= % /. PolyGamma[0,1+x_] -> PolyGamma[0,x] + (1/x)

s 4 s
4 + s PolyGamma[0, -] - s (- + PolyGamma[0, -])
4 s 4
Out[6]= -----------------------------------------------
2

In[7]:= Expand[%]

Out[7]= 0

(* So the result desired below appears. *)

Steve Christensen
MathSolutions, Inc.