# Bessel Function order 1

1. Apr 7, 2009

### salla2

Hi everyone, I need some help solving a bessel function of the 1st order. The equation is used to calculate the mutual inductance between two inductors. The equation is:

M=(1.45*10^-8)*integral [J1(1.36x)J1(0.735x)exp(-13.6x)]dx

the integral is from zero to infinity.

Thank you.

2. Apr 9, 2009

### Thaakisfox

Generally for Bessel-functions of order $$\nu$$ we have the following:

$$\int_0^{\infty}J_{\nu}(ax)J_{\nu}(bx)e^{-cx}dx = \frac{1}{\pi\sqrt{ab}}Q_{\nu - \frac12}\left(\frac{a^2+b^2+c^2}{2ab}\right)$$

Here Q_n(x) are legendre functions of the second kind.

For more details check "Watson: A treatise on the theory of Bessel functions" If you need something considering Bessel functions, then its in this book... :D

3. Apr 9, 2009

### coomast

The answer is the same... it is not an easy question, if you need to understand the answer, look into the book given by Thaakisfox. It is considered one of the most complete and advanced books on Bessel functions. It will take up a lot of time to read. I use it for looking things up.

coomast

Last edited by a moderator: Apr 24, 2017
4. Apr 10, 2009

### salla2

J1(x)=(x/2)[1-(((x/2)^2)/(2*!^2))+(((x/2)^4)/(3*2!^2))-.....]

I substituted x with 1.36x for the 1st term in the equation above and 0.735x for the 2nd term in the equation above, then plugged the value of J1(1.36) and J1(0.735x) in

M=(1.45*10^-8)*integral [J1(1.36x)J1(0.735x)exp(-13.6x)]dx

and calculated the value of M from zero to infinity using a TI89 calculator, I was able to obtain the value for M as 2.76585x10^-12

Is this process correct? I also got the book Thaakisfox mentioned from the library today and I'm working to see if there's an alternative solution that I can actually understand. Thank you guys for your help, I appreciate it.

salla2

5. Apr 10, 2009

### coomast

Hello salla2,

I did this using the software program maxima and got the following output:

$$2.82623 \cdot 10^{-12}$$

I assume this will be the same value as the one you have if you would use more terms. Attached is a picture of the maxima console. It is best to use numerical techniques for this kind of integrals especially if you are interested in a numerical value and not as such in a formula. The formula's are often very difficult to handle practically. This does not mean that they do not have any meaning, they are important for theoretical investigations.

best regards,

coomast

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6. May 10, 2009