# [SciPy-User] characteristic functions of probability distributions

David Goldsmith d.l.goldsmith@gmail....
Thu Nov 5 22:05:05 CST 2009

On Mon, Nov 2, 2009 at 12:51 PM, nicky van foreest <vanforeest@gmail.com>wrote:

> Hi Josef,
> > Second related question, since I'm not good with complex numbers.
> >
> > scipy.integrate.quad of a complex function returns the absolute value.
> > Is there a numerical integration function in scipy that returns the
> > complex integral or do I have to integrate the real and imaginary
> > parts separately?
>
> You want to compute \int_w^z f(t) dt? When f is analytic (i.e.,
> satisfies the Cauchy Riemann equations) this integral is path
> independent. Otherwise the path from w to z is of importance. You
> might like the book Visual Complex Analysis by Needham for intuition.
>

Furthermore, if f is analytic in an (open) region R homotopic to an (open)
disc, then the integral (an integer number of times) around *any* _closed_
path wholly in R is identically equal to zero; there's a similar statement
(though the end value is a multiple of 2ipi) if f has only poles of finite
order in R.  (Indeed, these properties should be used to unit test any
numerical complex path integration routine.)  Are any of your paths closed?

DG

>
> bye
>
> Nicky
>
> >
> > Thanks,
> >
> > Josef
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> >
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