Calculus Based Research
Art Credit: BLEACH Manga Edit by Soleim (Link)
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**Note: The contents of this page - except the formatting - are from when I first made this website before I started college. I have decided not to update the pictures with LaTeX documents in order to preserve my work and keep it authentic.**
Integration Technique(s)
Integration Technique(s)
Feynman's Technique
Feynman's Technique
A counterintuitive way to solve for integrals that are not solvable under elementary operations
As seen on the image on the right, the problem introduces the Feynman's technique on a seemingly ordinary function.
It also contains integration by parts, the gaussian integral, and differential equations.
Famous Integrals
Famous Integrals
Gaussian Integral
Gaussian Integral
One of the most prime examples of solving an integral without actually integrating.
Clever manipulating results in a perfect way to break through a problem almost seeming impossible.
The most difficult part arises from finding an initial function that helps achieve our goal, especially to incorporate the Feynman's Technique. The rest is elementary.
Pi and Gamma Functions
Pi and Gamma Functions
This proof by induction shows that Pi and Gamma functions are valid functions of factorials, due to the conditions enlisted:
Pi(0) = 0! = 1
Base Step
Pi(n) = n!
Induction Step
The same concept could be applied to the Gamma function, except the original conditions' input should be raised by 1, resulting in:
Gamma(1) = 0! = 1
Base Step
Gamma(n+1) = n!
Induction Step
Bose-Einstein Integral
Bose-Einstein Integral
This work associates the integral in that form with the Gamma and Reimann Zeta function.
We use the formula for the value at which a series converges:
r/1-r
This helps us finally reach the Bose-Einstein integral we are looking for.
This is seen in the Ramanujan summation section, where his summation ends with this integral when x = 1.
General Proofs
General Proofs
Power Rule
Power Rule
Simple Case:
This is a very simple proof of the power rule, as used for all functions in the form x^n.
The exponent rule of logarithms makes this proof very basic, as the exponent becomes a constant based on that rule.
Generalized Case:
This case incorporates the chain rule and expands the power rule's grasp to all functions f(x)^n, where f(x) is any differentiable function.
Derivative of ln(x)
Derivative of ln(x)
Implicit Differentiation
The whole proof is based on switching the equation to an exponential, rather than a logarithmic. The beauty of e^x's derivative being itself shines through.
Limit definition of a derivative
Since e can be described as a limit, the entire proof turns elementary.
Logarithm rule of exponents plays a key role in this proof as well.
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