Accursed Bessel Functions!

Accursed Bessel Functions!

Post by Tim Wescot » Fri, 02 Apr 2010 04:24:49

OK: I'm posting this because I'm lazy. I haven't even _tried_ to look
this up yet, or to attempt to *** an answer out of Maxima. If there
_is_ an answer, I'm sure it'll be a Bessel function, and those things
always hit me like headlights do a deer.

So, anyone feel like some math hand-holding? Here's the integral I'm
trying to find an answer for:

integral from -pi to pi, e^(-a + b * cos(theta)) d theta,

a > b > 0.

This seems to be the class of problem that I can answer by myself, but
only after I've beat my head against the wall for ages or held myself
out for public ridicule -- I'm in a hurry, so I'm going to embarrass
myself _first_, in hopes of finding the solution quicker.

Tim Wescott
Control system and signal processing consulting

Accursed Bessel Functions!

Post by Tim Wescot » Fri, 02 Apr 2010 04:35:14

Worse, maybe it's _not_ covered by Bessel functions, and I'm _really_ on
my own. Argh.

(Proceeding to plug away at the math).

Tim Wescott
Control system and signal processing consulting


Accursed Bessel Functions!

Post by glen herrm » Fri, 02 Apr 2010 04:52:45

Well, first I would factor out exp(-a), as that is constant.

Next I put it into

which says that "Mathematica could not find a formula for your

But that only does indefinite integrals, and this seems like a
case where the definite integral might be possible to evaluate
even though the indefinte integral doesn't have an analytical
solution. (Not that you can find a numeric solution for an
indefinite integral.)

-- glen

Accursed Bessel Functions!

Post by Jerry Avin » Fri, 02 Apr 2010 04:53:04

I'm too rusty to see why iterated integration by parts won't work.

Discovery consists of seeing what everybody has seen, and thinking what
nobody has thought. .. Albert Szent-Gyorgi

Accursed Bessel Functions!

Post by dvsarwat » Fri, 02 Apr 2010 05:14:21

Yes, it is a Bessel function (times a constant).
Look at Abramowitz and Stegun, 9.6.16 which says

I_0(z) = (1/pi) integral from 0 to pi e^(z cos theta) d theta

where I_0(z) is a modified Bessel function of the first kind
and order 0. Shows up a lot in studies of noncoherent
FSK, DPSK and the like and in the Ricean pdf which
is the pdf of the amplitude of a sinusoid plus narrowband
Gaussian noise.

Hope this helps.

--Dilip Sarwate

Accursed Bessel Functions!

Post by Michael Pl » Fri, 02 Apr 2010 08:26:01

Well, ignoring the exp(-a) as Glen did, I put it in Mathematica and

Integrate[Exp[b*Cos[th]], {th, -pi, pi}]

(where I've typed out "pi" for your newsreader's sake)

And also ran:

FullSimplify[%, Im[b] == 0 && Re[b] > 0]

The result was:

2 pi (BesselI[0, b] - StruveL[0, b])

(...I'll post again if I see any font problems in google...)


Accursed Bessel Functions!

Post by JCH » Fri, 02 Apr 2010 15:12:05

"glen herrmannsfeldt" < XXXX@XXXXX.COM > schrieb im Newsbeitrag

For definite integral use Gaussian Integration:



Regards JCH


Accursed Bessel Functions!

Post by Scott Hemp » Mon, 05 Apr 2010 11:33:05

Tim Wescott < XXXX@XXXXX.COM > writes:

Mathematica 7.0:

In[1]:= Integrate[E^(-a+b*Cos[theta]),{theta,-Pi,Pi},Assumptions->a>b>0]

2 Pi BesselI[0, b]
Out[1]= ------------------

Scott Hemphill XXXX@XXXXX.COM
"This isn't flying. This is falling, with style." -- Buzz Lightyear