# Math Help - Please help derivation of Bessel Function

1. ## Please help derivation of this integration

I am studying Bessel Function in my antenna theory book, it said:
$\pi j^n J_n(z)=\int_0^{\pi} \cos(n\phi)e^{+jz\cos\phi}d\phi$

I understand:
$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi-m\theta)} d\theta$
Can you show me how do I get to
$\pi j^m J_m(z)=\int_0^{\pi} \cos(m\theta)e^{+jz\cos\theta}d\theta$

I tried $e^{jm\theta}=\cos m\theta +j\sin m \theta$ but it is not easy. Please help.

2. ## Re: Please help derivation of this integration

Originally Posted by Alan0354
I am studying Bessel Function in my antenna theory book, it said:
$\pi j^n J_n(z)=\int_0^{\pi} \cos(n\phi)e^{+jz\cos\phi}d\phi$

I understand:
$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi-m\theta)} d\theta$
Can you show me how do I get to
$\pi j^m J_m(z)=\int_0^{\pi} \cos(m\theta)e^{+jz\cos\theta}d\theta$

I tried $e^{jm\theta}=\cos m\theta +j\sin m \theta$ but it is not easy. Please help.
You start with:

$J_m(z)= \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin(\phi)-m\phi)} d\phi = \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos(\phi + \pi/2)-m\phi)} d\phi$

Now change the variable to $\xi=\phi+\pi/2$, then as before use:

$e^{jm\xi}=\cos (m\xi) +j \sin (m \xi)$

and use the symmetry properties ( and periodicity? )of the functions involved ...

.

3. ## Re: Please help derivation of this integration

Originally Posted by zzephod
You start with:

$J_m(z)= \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin(\phi)-m\phi)} d\phi = \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos(\phi + \pi/2)-m\phi)} d\phi$

Now change the variable to $\xi=\phi+\pi/2$, then as before use:

$e^{jm\xi}=\cos (m\xi) +j \sin (m \xi)$

and use the symmetry properties ( and periodicity? )of the functions involved ...

.
Thanks for you time.

Is it supposed to be
$J_m(z)= \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin(\phi)-m\phi)} d\phi = \frac{1}{2\pi}\int_0^{2\pi}e^{j[z\cos(\phi + \pi/2)-m(\frac{\pi}{2}+\phi)]} d\phi$

I tried something similar but with no luck, can you show more steps?

Thanks

4. ## Re: Please help derivation of this integration

Originally Posted by Alan0354
Thanks for you time.

Is it supposed to be
$J_m(z)= \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin(\phi)-m\phi)} d\phi = \frac{1}{2\pi}\int_0^{2\pi}e^{j[z\cos(\phi + \pi/2)-m(\frac{\pi}{2}+\phi)]} d\phi$

I tried something similar but with no luck, can you show more steps?

Thanks
No I am just using the identity $\sin(\phi)=\cos(\phi-\pi/2)$ (note the wrong sign in the erlier post) to convert the sine to a cosine. Then the change of variable will be $\xi=\phi-\pi/2$

.

5. ## Re: Please help derivation of this integration

Originally Posted by zzephod
No I am just using the identity $\sin(\phi)=\cos(\phi-\pi/2)$ (note the wrong sign in the erlier post) to convert the sine to a cosine. Then the change of variable will be $\xi=\phi-\pi/2$
$J_m(z)= \frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos(\phi-\frac{\pi}{2})-m\phi)} d\phi$
$u=\phi-\frac{\pi}{2}\;\Rightarrow\;\phi=u+\frac{\pi}{2},\ ;du=d\phi$
$= \frac{1}{2\pi}\int_{-\frac{\pi}{2}}^{\frac{3\pi}{2}}e^{j[z\cos(u)-m(u+\frac{\pi}{2})]} du$

So what is the next step?

Thanks

6. ## Re: Please help derivation of this integration

$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi-m\theta)} d\theta=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi) }[\cos (m\theta)-j\sin (m\theta)]d\theta$

$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi)} \cos (m\theta)d\theta -\frac{j}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi)}\sin (m\theta)d\theta$

$J_m(z)$ is real.

$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi)} \cos (m\theta)d\theta$

But I still don't get the $\frac{1}{\pi j^m}$ part of it and the interval of [0, $\pi$].

7. ## Re: Please help derivation of this integration

Originally Posted by Alan0354
$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\phi-m\theta)} d\theta$
One mistake that seems to be following you...there is no $\phi$ in this equation. The function is defined as
$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\theta-m\theta)} d\theta$

-Dan

8. ## Re: Please help derivation of this integration

Originally Posted by topsquark
One mistake that seems to be following you...there is no $\phi$ in this equation. The function is defined as
$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\theta-m\theta)} d\theta$

-Dan
Sorry, it's a typo. But you know what I meant.

9. ## Re: Please help derivation of Bessel Function

I worked on a few more steps:

$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\theta-m\theta)} d\theta=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos u-m\frac{\pi}{2}+mu)} du$

$=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos u+mu)}e^{-jm\frac{\pi}{2}} \;du=-\frac{j\sin(m\frac{\pi}{2})}{2\pi}\int_0^{2\pi}e^{ j(z\cos u+mu)}du$

With this, $J_m(z)=0$ for m=even. That is not going to get the answer. Also, how do you change $\;\int_0^{2\pi}e^{j(z\cos u+mu)}du\;$ to $\;2 \int_0^{\pi}e^{j(z\sin\theta-m\theta)}d\theta$?

10. ## Re: Please help derivation of Bessel Function

Originally Posted by Alan0354
I worked on a few more steps:

$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\theta-m\theta)} d\theta=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos u-m\frac{\pi}{2}+mu)} du$

$=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos u+mu)}e^{-jm\frac{\pi}{2}} \;du=-\frac{j\sin(m\frac{\pi}{2})}{2\pi}\int_0^{2\pi}e^{ j(z\cos u+mu)}du$

With this, $J_m(z)=0$ for m=even. That is not going to get the answer. Also, how do you change $\;\int_0^{2\pi}e^{j(z\cos u+mu)}du\;$ to $\;2 \int_0^{\pi}e^{j(z\sin\theta-m\theta)}d\theta$?
I was wrong in the last step, it should be how do I get to $\int_0^{\pi}e^{jz\sin\theta}\cos m\theta d\theta$

11. ## Re: Please help derivation of Bessel Function

Hi Zzephod , You are right. I since work on a few more steps

$J_m(z)=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\sin\theta-m\theta)} d\theta= \frac{1}{2\pi}\int_0^{2\pi} e^{j(z\cos u-m\frac{\pi}{2}+mu)} du=\frac{1}{2\pi}\int_0^{2\pi}e^{j(z\cos u+mu)}e^{-jm\frac{\pi}{2}} \;du$

$J_m(z)=\frac{e^{-jm\frac{\pi}{2}}}{2\pi}\int_0^{2\pi}e^{jz\cos u}[\cos(m\theta)+j\sin(m\theta)]d\theta$

As for $e^{-j(\frac{m\pi}{2})}=\cos\frac{m\pi}{2}-j\sin\frac{m\pi}{2}$

For m=odd, $\;\cos\frac{m\pi}{2}=0\;$ and $\;-j\sin\frac{m\pi}{2}=j^{-m}$

For m=even, $\;j\sin\frac{m\pi}{2}=0$

For m=0 $\Rightarrow\;\cos\frac{m\pi}{2}=1,\;\;$ m=2 $\Rightarrow\;\cos\frac{m\pi}{2}=-1,\;\;$ m=4 $\Rightarrow\;\cos\frac{m\pi}{2}=-1,\;\;$.......Therefore $\Rightarrow\;\cos\frac{m\pi}{2}=j^m,\;\;$ not $\;j^{-m};$

What did I do wrong this time?

12. ## Re: Please help derivation of Bessel Function

Stupidity strikes again!!! I just realize $e^{j(\frac{m\pi}{2}=j^m=j{-m}$!!!! So

$J_m(z)=\frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos u}\cos(m\theta)d\theta+ \frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos u}j\sin(m\theta)d\theta$

13. ## Re: Please help derivation of Bessel Function

Originally Posted by Alan0354
.....Therefore $\Rightarrow\;\cos\frac{m\pi}{2}=j^m,\;\;$ not $\;j^{-m};$

What did I do wrong this time?
Nothing, $\cos(x)$ is symmetric so $\cos(x)=\cos(-x)$.

Or more explicitly

$\displaystyle j^{-m}=\frac{1}{j^m}=\frac{j^m}{j^{2m}}=(-1)^m j^m=j^m$ for $m$ even

14. ## Re: Please help derivation of Bessel Function

$J_m(z)=\frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos \theta}\cos(m\theta)d\theta+ \frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos \theta}j\sin(m\theta)d\theta$

How do I proof

$\frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos \theta}\cos(m\theta)d\theta\;\neq\;0$

AND

$\frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos \theta}j\sin(m\theta)d\theta=0$

Thanks

15. ## Re: Please help derivation of Bessel Function

There is no easy answer or easy way to integrate $\;\frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos \theta}\cos(m\theta)d\theta\;$ AND $\;\frac{j^{-m}}{2\pi}\int_0^{2\pi}e^{jz\cos \theta}j\sin(m\theta)d\theta$

I tried $\;e^{jz\cos \theta}=\cos (z\cos \theta) +j\sin(z\cos \theta)\;$ Where you need to solve $\;\int_0^{2\pi}[\cos (z\cos \theta) +j\sin(z\cos \theta)][\cos(m\theta)+j\sin(m\theta)] d\theta\;$

I tried $\;e^{jz\cos \theta}=\sum_0^{\infty}\frac {(jz\cos\theta)^k}{k!}\;$ Where you need to solve $\;\int_0^{2\pi}\sum_0^{\infty}\frac {(jz\cos\theta)^k}{k!}[\cos(m\theta)+j\sin(m\theta)]d\theta$.

Please help

Thanks