# Accursed Bessel Functions!

Started by March 31, 2010
```OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to look
this up yet, or to attempt to choke 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
www.wescottdesign.com
```
```Tim Wescott wrote:
> OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to look
> this up yet, or to attempt to choke 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.
>
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
www.wescottdesign.com
```
```Tim Wescott <tim@seemywebsite.now> wrote:

> 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.

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

Next I put it into http://integrals.wolfram.com/

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

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
```
```Tim Wescott wrote:
> OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to look
> this up yet, or to attempt to choke 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.

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

Jerry
--
Discovery consists of seeing what everybody has seen, and thinking what
nobody has thought.    .. Albert Szent-Gyorgi
&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;&#4294967295;
```
```On Mar 31, 2:24&#4294967295;pm, Tim Wescott <t...@seemywebsite.now> wrote:

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

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

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
```
```Glen wrote:
>Tim Wescott <tim@seemywebsite.now> wrote:
>
>> 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.
>
>Well, first I would factor out exp(-a), as that is constant.
>
>Next I put it into http://integrals.wolfram.com/
>
>which says that "Mathematica could not find a formula for your
>integral."
>
>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.)

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

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

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...)

HTH,
Michael

```
```"glen herrmannsfeldt" <gah@ugcs.caltech.edu> schrieb im Newsbeitrag
news:hp096e\$blq\$4@naig.caltech.edu...
> Tim Wescott <tim@seemywebsite.now> wrote:
>
>> 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.
>
> Well, first I would factor out exp(-a), as that is constant.
>
> Next I put it into http://integrals.wolfram.com/
>
> which says that "Mathematica could not find a formula for your
> integral."
>
> 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.)
>

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

For definite integral use Gaussian Integration:

* http://home.arcor.de/janch/janch/_news/20100401-numerical-integration/

EXAMPLE: Page 1
PROGRAM: Page 2

--
Regards JCH

u
> -- glen

```
```Tim Wescott <tim@seemywebsite.now> writes:

> OK:  I'm posting this because I'm lazy.  I haven't even _tried_ to
> look this up yet, or to attempt to choke 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.

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]= ------------------
a
E

Scott
--
Scott Hemphill	hemphill@alumni.caltech.edu
"This isn't flying.  This is falling, with style."  -- Buzz Lightyear
```