This section is dedicated to the pleasure of calculating integrals. Have a look at some integrals I find most interesting.

*Arctangent’s in the Spotlight*

- where is Catalan’s constant.
- For any and ,
- For any ,
- For any for which ,
- For any with ,
- For any for which and ,
- For any for which and ,
- For any with ,
- For any ,

*For the love of Euler’s Constant*

- Suppose is an complex matrix whose real part is positive definite. If denotes the standard complex inner product then where the principal value of the square root is taken. (See
*A Brief Introduction to Theta Functions*by Richard E. Bellman) - For any ,
- For any ,
- For any ,
- For any ,
- For any and ,

*Logs. Everywhere.*

- For any ,
- For ,

*Radical Integrals*

- Let be a real-valued differentiable function on that is non-negative and monotonically increasing. Then
- Let be open. Suppose is a continuously differentiable function that is monotone increasing and onto. Then where denotes Catalan’s constant
- Let with . Then,
- Consider the following functions for : and If denotes the unique solution to the equation over , then
- Let and real satisfy . Consider the function for where is the incomplete gamma function with integral representation Then, with and

*Jus’ be Cos…*

- For any ,
- For any ,
- For any and , where
- For any for which ,
- For any with and ,
- Let be a real sequence consisting of , nonzero, distinct terms. Let and be given. Then where
- For any with , where

*Spot the *

- For any ,
- For any ,
- Let and define by . If satisfy , then

*Some Building Blocks*

- For any and natural ,
- For any and ,
- For any and ,
- For any and ,
- For any and ,
- For any such that , and ,
- For any with
- For any and ,
- For any such that , where denotes the fractional part of .
- Let satisfy and . Define a finite set of polynomials Suppose we have a finite set of real-valued continuous functions on , and let, for any , Then,

Moreover,

#### Who Said Infinity Can’t Be Fun?

- For any constants , and function for which , and for almost every , there holds
- Let be a real sequence consisting of , nonzero, distinct terms. Let functions satisfy, for each , Then where for .
- Let with and . Then,
- For any , with and
- For any and natural such that and ,
- For any for which and ,
- For any ,
- For any and ,
- For any ,
- Let denote the zeroth order Bessel function of the firtst kind. Then, for any ,

#### Complex Analysis to the Rescue!

- For where , and and where and
- For any and such that and , where and, for , we define
- Given where and , for all there holds Consequently, there holds for all and
- Given where and , for all there holds where and
- For any and with ,
- For any ,
- For any and with ,

Copyright © 2020 Yohance Osborne