Latest update

- Hosting a Remote Node
- JSON-RPC Not Working as Shown in Wallet Guide
- Altcoins to Monero payment buttons for website?
- Problem of deploying contract with “testRPC” and “remix” : “creation of contract pending …”
- Best Method to Increase TPS on PoA Blockchain
- “The type of the local variable x is uint[] storage,but since storage is not dynamically allocated.” What exactly it means?
- Error: VM Exception while processing transaction: revert when contract function calling contract function
- How can I transfer ownership to another contract?
- “The transaction execution will likely fail. Do you want to force sending? VM Exception while processing transaction: out of gas”
- Result of calling web3.version.getNetwork() from browser-side code is always undefined?
- Would I be infringing on a trademark if my company name is later trademarked?
- Does keeping an MD5 hash of user data violate GDPR?
- Is the term “race” defined by Public Law enacted by Congress of the United States
- Can a Guardian (parent) force a 17 year old to move out of their house in Colorado?
- Speed fine in Wyoming over 30mph - What should I do now?
- двоеточие перед пояснительными словами когда можно ставить?
- Where can I get professional medical advice online?
- Why isn't Tylenol safe?
- Preventing hair loss
- Trying to get rid of a habit in an environment which is hindering the resistance

# 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!