integral of exp(-x/a)*sinh(sqrt(b*x)) from 0 to infinity vanished?

asked 2023-11-04 10:37:59 +0200

KazuFurutaka gravatar image

Hi there.

I've tried to calculate integral, $$\int_{0}^{\infty} \exp{(-x/a)}\:\sinh{(\sqrt{bx})}\:dx,$$

a, b = var("a b")
Watt = exp(-x/a)*sinh(sqrt(b*x))
defInt = integrate(Watt,x,0,infinity)

For a>0 and b>0, the result seems to be $$\frac{\sqrt{\pi}\:a^{3/2}\:\sqrt{b}\:e^{ab/4}}{2},$$ and actually

defInt_a1_b1 = integrate(Watt(a=1,b=1),x,0,infinity)

yields $$\frac{1}{2}\sqrt{\pi}\:e^{\frac{1}{4}}.$$


defInt = integrate(Watt,x,0,infinity)

vanishes... $$0$$

Where did I make mistakes?

Thanks in advance,


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some issue in maxima. you can try other engines:

sage: integrate(Watt,x,0,infinity,algorithm='sympy')
cases(((abs(arg(a)) < 1/2*pi, 1/2*sqrt(pi)*a^(3/2)*sqrt(b)*e^(1/4*a*b)), (1, integrate(e^(-x/a)*sinh(sqrt(b*x)), x, 0, +Infinity))))
FrédéricC gravatar imageFrédéricC ( 2023-11-04 20:44:05 +0200 )edit


sage: mathematica.Integrate(Watt, [x, 0, oo])
ConditionalExpression[(a^(3/2)*Sqrt[b]*E^((a*b)/4)*Sqrt[Pi])/2, Re[a] > 0]
sage: mathematica.Refine(mathematica.Integrate(Watt, [x, 0, oo]), mathematica.And(a>0, b>0)).sage()
Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2023-11-04 21:11:42 +0200 )edit

@FrédéricC :

  • I took the liberty of refoprmatting your comment, which you had formatted in markdown ;-)...

  • You should make an answer of your comment, for the benefiot of future (per-)users...


Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2023-11-04 21:15:10 +0200 )edit

Thanks @FrédéricC for your comment. Indeed,


yields 0.

But it seems to me that this is not due to Maxima itself because direct run of Maxima from console (and via Jupyterlab Desktop) yields the expected result, $\frac{\sqrt{\pi}a^{3/2}\sqrt{b}e^{ab/4}}{2}$.

I'll try other engines...

KazuFurutaka gravatar imageKazuFurutaka ( 2023-11-05 02:06:30 +0200 )edit

@Emmanuel Charpentier, thanks for your comment. Do I need to install extra/optional packages to do the following?:

sage: mathematica.Integrate(Watt, [x, 0, oo])

When I did that I got (in short):

TypeError: unable to start mathematica: End Of File (EOF). Exception style platform.
Mathematica finished running /usr/bin/sh -c stty -echo; math -rawterm
command: /usr/bin/sh
args: ['/usr/bin/sh', '-c', 'stty -echo; math -rawterm']
buffer (last 100 chars): b''
before (last 100 chars): b'/usr/bin/sh: line 1: math: command not found\r\n'
after: <class 'pexpect.exceptions.EOF'>
match: None
match_index: None

By the way I use SageMath 10.1 (from console as well as Jupyterlab Desktop) built from source, on Fedora linux 38 (x86_64).

KazuFurutaka gravatar imageKazuFurutaka ( 2023-11-05 03:53:40 +0200 )edit