Gamma function

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The first two are merely definitions of a function. There is no nice closed form for inverting the Gamma function, but there are papers out there on how to compute/approximate the inverse.

The expression on the left hand side of the last line is equal to pi·csc(pi·x), where csc is the cosecant. If I'm not confused, it only takes values between 1/2 and 1, monotonically growing, and for all other values they are mere periodic repetitions of those. And At 1/2 it's pi, which is larger than \sqrt(3·\sqrt(3))/2, so there's no solution.

>>8570929
[math]x_{\pm}^{(n)} = 0.318\left( \pm 1.672 i + \pi \left( 2n + \frac{1}{2} \right) \right)[/math]
There's probably a really smart way to do this, good luck.

>>8570936
I thought at 1/2 it's sqrt(pi)/2 no?

>>8570929
Gamma function is memoryless

>>8570929
Maybe use Euler's reflection formula...
Gamma(z) Gamma(1-z) = pi/ sin(pi z) when z is not an integer.

So pi/sin(pi z) = sqrt(sqrt(3))/2

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>>8570929
[math]\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin{\pi z}}[/math]
[math]\sin{\pi z}=A, A = \frac{2\pi}{\sqrt{3\sqrt{3}}}[/math]
[math]\pi z = x+iy[/math]
[math]\sin{x}\cosh{y}+i\sinh{y}\cos{x}=A[/math]
[math]x=\pi/2+2\pi n[/math]
[math]\cosh{y}= A[/math]
[math]y = arcosh(A)[/math]
[math]z=\frac{1}{2}+2n+i\frac{1}{\pi}\ln{(A\pm \sqrt{A^2-1})}[/math]

Habby new yearz :DD

From 2, [math]x[/math] must be a natural integer since the factorial is defined for natural integers.
Therefore [math]\Gamma \,\left( 1 \,-\, x \right) \,\Gamma \left( x \right)[/math] is an integer since it equals a factorial. However [math]\frac{\sqrt{3 \,\sqrt{3}}}{2}[/math] is not an integer.
There is no solution.

>>8571072
Are you actually retarded? Do you even know what the gamma function is?

>>8571007
this, just write it in terms of sin(x)

>>8570951
Gamma would be sqrt pi, that expression with the two gammas gives pi.

>>8570936
What papers senpai?

>>8571007
>[/math][math]?

>>8571078
The [math]\Gamma[/math] function is a complex analytical function defined on [math]\mathbf C \,\setminus\, -\mathbf Z[/math] for which the application [math]z \,\longmapsto\, \Gamma \left( z \,-\, 1 \right)[/math] extends the factorial.

This does not mean the factorial is defined on [math]\mathbf C \,\setminus\, -\mathbf Z^*[/math], Pajeet.

>>8571248
There's a StackExchange thread about someone trying to invert the Gamma function out there, and there I saw it being linked