Latest update

- I-539 Extension of Stay Application from my last visit is still pending. Can I re-visit the U.S.?
- Upper East Side to I-95 South via Car
- Virtually 'drive' a route with Google Sreetview
- Can I use an ESTA of an expired Italian passport with my Brazilian passaport?
- How do stained bedclothes affect the nightly price?
- What plant is this seed of?
- Is Nietzsche an Ethical Egoist?
- McGee's Counterexample to Modus Ponens
- Debate about what degree of Tzedakah this is
- Appoggiatura chord
- Structural Navigation shows a wrong link
- 2005 Ford E150 Econoline
- Why are car fluid suction pumps not popular with mechanics (for oil changes)?
- Why do vertical images flip horizontal after upload?
- Is _access: 'TRUE' mandatory for a public route?
- Does Social Security have a $2.5T surplus?
- Help with this question about deferred perpetuity question!!
- Automate the Boring Stuff with Python - The Collatz Sequence Project
- What did the Merovingian expect to happen when he told his goons to kill Neo?
- How many aurors guarded Hogwarts in Harry Potter's 6th year?

# Zeroth order modified Bessel function integral representation

2018-06-19 23:57:52

I'm trying to understand the derivation of:

$$ I_0(x) = \frac{1}{\pi}\int_{0}^{\pi} \exp(x\cos\theta) \, d\theta$$

I'm trying to use this generating function:

$$ \exp\left(\frac{x}{2}(z-z^{-1})\right) = \exp(x\cos\theta) = \sum_{n=-\infty}^\infty I_n(x) \exp (in\varphi)$$

Is this correct? (left side real and right side complex). Then, i'm using:

$$ \int_0^\pi \exp(x\cos\theta) \,d\theta = \frac 1 \pi \int_0^\pi \sum_{n=-\infty}^\infty I_n(x) \exp (in\varphi) \, d\theta$$

But this seems no quite right.

Thanks!