Gamma function

This is gamma function.

Z include complex numbers. Z>0

∴0!=1

Then, you see Riemann zeta function.

You can also write this.

You can get 1 finally.

Fractional Calculus

Irrationality is a philosophical question. Now we talk about AI, but the computational world often avoids this problem. If D is prime numbers, this square root of D is irrational. When you have 100=10*10, 10 is not the prime numbers. 7 is the biggest prime number in 10. The square root of 7 is irrational. 2,3,5,7 are prime numbers in 100. Then you expand integers eternally. I define that this is fractal.

This is called Cauchy’s formula for repeated integration. The generalization of the factorial function is the gamma function.

This is the left Riemann-Liouville integral. α is complex and greater than zero. The special case of α = 1/2 is called the semi-integral.

The gamma function is not defined for zero or for negative integers.

This is irrational.

This is the left Riemann-Liouville fractional derivative.

The Riemann-Liouville differintegral continuously transforms between the functions f(x) = x, f(x) = 1, and f(x) = (1/2)x^2, and α ranging from -1 to 1.

This is waving.