Differentiating under integral

Prove: (Euler Gamma Γ Function)
$latex \displaystyle n! = \int_{0}^{\infty}{x^{n}.e^{-x}dx}$

Proof:
∀ a>0
Integrate by parts:

$latex \displaystyle\int_{0}^{\infty}{e^{-ax}dx}=-\frac{1}{a}e^{-ax}\Bigr|_{0}^{\infty}=\frac{1}{a}$

∀ a>0
$latex \displaystyle\int_{0}^{\infty}{e^{-ax}dx}=\frac{1}{a}$ ...[1]

Feynman trick: differentiating under integral => d/da left side of [1]

$latex \displaystyle\frac{d}{da}\displaystyle\int_{0}^{\infty}e^{-ax}dx= \int_{0}^{\infty}\frac{d}{da}(e^{-ax})dx=\int_{0}^{\infty} -xe^{-ax}dx$

Differentiate the right side of [1]:
$latex \displaystyle\frac{d}{da}(\frac{1}{a}) = -\frac{1}{a^2}$
=>
$latex a^{-2}=\int_{0}^{\infty}xe^{-ax}dx$

Continue to differentiate with respect to 'a':
$latex -2a^{-3} =\int_{0}^{\infty}-x^{2}e^{-ax}dx$
$latex 2a^{-3} =\int_{0}^{\infty}x^{2}e^{-ax}dx$
$latex \frac{d}{da} \text{ both sides}$
$latex 2.3a^{-4} =\int_{0}^{\infty}x^{3}e^{-ax}dx$
...
...
$latex 2.3.4\dots n.a^{-(n+1)} =\int_{0}^{\infty}x^{n}e^{-ax}dx$
Set a = 1
$latex \boxed{n!=\int_{0}^{\infty}x^{n}e^{-x}dx}$ [QED]

Another Example using "Feynman Integration":

$latex \displaystyle \text{Evaluate }\int_{0}^{1}\frac{x^{2}-1}{ln x} dx$

$latex \displaystyle \text{Let I(b)} = \int_{0}^{1}\frac{x^{b}-1}{ln x} dx$ ; for b > -1

$latex \displaystyle \text{I'(b)} = \frac{d}{db}\int_{0}^{1}\frac{x^{b}-1}{ln x} dx = \int_{0}^{1}\frac{d}{db}(\frac{x^{b}-1}{ln x}) dx$

$latex x^{b} = e^{ln x^{b}} = e^{b.ln x} $

$latex \frac{d}{db}(x^{b}) = \frac{d}{db}e^{b.ln x}=e^{b.ln x}.{ln x}= e^{ln x^{b}}.{ln x}=x^{b}.{ln x}$

$latex \text{I'(b)}=\int_{0}^{1} x^{b} dx=\frac{x^{b+1}}{b+1}\Bigr|_{0}^{1} = \frac{1}{b+1}$
=>
$latex \text {I(b)} = ln (b+1) + C$

Let b=0
I(0) = 0= ln (1) + C = 0+C => C=0

$latex \boxed{I(b)=ln(b+1)}$

Let b= 2
$latex \displaystyle\int_{0}^{1}\frac{x^{2}-1}{ln x} dx = I(2) = ln (3)$ [QED]

Visualize e^iπ = -1

Visualize by mapping Exp e (red and pink arrows):

$latex e^{i\pi} = -1$

Euler on Math Education

Euler was invited by Peter I of Russia in 1727 to work in the
Petersburg Academy of Sciences. He introduced the fundamental math
disciplines in school math education:
1. Arithmetic
2. Geometry
3. Trigonometry
4. Algebra
these 4 are taught as separate and specific subjects, versus 19 duplicated disciplines in Europe.

Euler influenced not only in Russia schools, but in schools worldwide today.

Source: Russian Mathematics Education
Vol 1: History and world significance
Vol 2: Programs and practices
(Publisher: World Scientific)

Euler: V- E + R = 2

1. Euler wrote to Goldbach @1750, "it astonishes me these properties have not been noticed by anyone else."

Euler's Polyhedron formula:

V- E + R = 2

Remember trick: “VERsion 2”
V= Vertices, E= Edges, R= Regions (or Faces)
Note: = 0 (instead of 2) for torus doughnut

2. Euler proved it 1 year later: intuition leads to discovery, then prove it by logic.

Example:
Football: Vertices=60, Edges E=90, Region R=32 faces (12 pentagons, 20 hexagons):
V- E + R = 60 - 90 + 32 = 2

[caption id="" align="alignright" width="300"] A football (or soccer ball) icon. (Photo credit: Wikipedia)[/caption]

Prime and Perfect Square

For all primes p ≠2, (a,b ∈Z)

p= a² + b² <=> p ≡ 1 mod 4


(2=1² + 1²)
5= 1² + 2² = 1 + 4 ≡ 1 mod 4
13= 2² + 3² = 4 + 9 ≡ 1 mod 4
17=1² + 4² = 1 + 16 ≡ 1 mod 4
29= 2² + 5² = 4 + 25 ≡ 1 mod 4
37= 1² + 6² = 1 +36 ≡ 1 mod 4

Notes:

1) Perfect squares (4, 9, 16, 25... ) ≡ 0 or ≡ 1 mod 4
2) Prime (4n+1) = a² + b²   (Euler took 7 yrs to prove)

3) Gauss expanded the proof to quadratic reciprocity (2 prime numbers p & q are linked by mod 4)