## #2732 wrong answer for similar to gaussian integral

None
closed
None
5
2014-06-04
2014-05-17
dan hayes
No

wxMaxima version: 13.4.0
Maxima version: 5.31.1
Maxima build date: 2013-09-24 09:49:12
Host type: i686-pc-mingw32
Lisp implementation type: GNU Common Lisp (GCL)
Lisp implementation version: GCL 2.6.8

```(assume(a>0,b>0), t1:integrate(exp(-a^2*x^2-b^2/x^2),x,0,inf), t2:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,0)
, t3:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,inf),ldisp([t1,t2,t3]));
```

By symmetry of x and -x the first two results, t1 and t2 , should be same which they correctly are. But then obviously the 3rd result, t3, should just be twice t1 (or t2) but instead it adds a term with the the positive exponent in the exponential to t1 which is incorrect. Also maxima should be able to do closed form integrals of the form `integrate(x^(2*i)*exp(-a*x^2-b/x^2),x,0,inf)` for any integer i by differentiating i times the closed form answer of

```integrate(exp(-a*x^2-b/x^2),x,0,inf)
```

with respect to a for i>0 or b for i<0 but it does not do so.

## Discussion

• Rupert Swarbrick - 2014-05-17

For reference, here's the missing transcript (with current git master):

```(%i1) assume(a>0, b>0);
(%o1)                           [a > 0, b > 0]
(%i2) t1:integrate(exp(-a^2*x^2-b^2/x^2),x,0,inf);
- 2 a b
sqrt(%pi) %e
(%o2)                         -------------------
2 a
(%i3) t2:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,0);
- 2 a b
sqrt(%pi) %e
(%o3)                         -------------------
2 a
(%i4) t3:integrate(exp(-a^2*x^2-b^2/x^2),x,-inf,inf);
2 a b               - 2 a b
sqrt(%pi) %e        sqrt(%pi) %e
(%o4)               ----------------- + -------------------
2 a                  2 a
(%i5) print([t1,t2,t3]);
- 2 a b              - 2 a b
sqrt(%pi) %e         sqrt(%pi) %e
[-------------------, -------------------,
2 a                  2 a
2 a b               - 2 a b
sqrt(%pi) %e        sqrt(%pi) %e
----------------- + -------------------]
2 a                  2 a
- 2 a b              - 2 a b
sqrt(%pi) %e         sqrt(%pi) %e
(%o5) [-------------------, -------------------,
2 a                  2 a
2 a b               - 2 a b
sqrt(%pi) %e        sqrt(%pi) %e
----------------- + -------------------]
2 a                  2 a
```

• dan hayes - 2014-05-24

Note the commercial version Macsyma gives the correct answer of
sqrt(%pi)exp(-2ab)/a for integrate(exp(-a^2x^2-b^2/x^2),x,-inf,inf);

I see that the source code for that now defunct project is made publicly available so perhaps those knowledgeable could use or find out how Macsyma did it.

• Dan Gildea - 2014-06-04
```src/defint.lisp:
o discontinuities-denom:
recursively traverse exp in order to find discontinuities such as
erfc(1/x-x) at x=0

tests/rtestint.mac:
o add integrate(exp(-x^2-1/x^2),x,-inf,inf);
```

#### Related

• Dan Gildea - 2014-06-04
• status: open --> closed
• assigned_to: Dan Gildea

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