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