Undergrad Closed Form for Complex Gamma Function

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The discussion revolves around finding a closed form expression for the complex gamma function, specifically ##\Gamma(\frac{1}{2}+ib)## where ##b## is a real number. A link to a MathOverflow post is shared as a potential resource for further information. Participants suggest browsing Wikipedia and Khan Academy for additional insights, while one notes that Stack Exchange can be overly complex. The integral representation of the gamma function is also mentioned as a foundational concept. The conversation emphasizes the need for accessible resources on this mathematical topic.
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Hi, @thatboi, Wikipedia could be worth browsing?. Personally, Stack Exchange is too...Complex :smile:,for me.
Some other suggestions: Khan Academy. You've found the solution. This is the path:

$$\Gamma (x)=\int_0^{\infty}t^{x-1}e^{-t}dt$$

PF, please check the LaTeX.

Love, peace
 

Thread 'Proving that convexity implies second order derivative being positive'

There are probably loads of proofs of this online, but I do not want to cheat. Here is my attempt: Convexity says that $$f(\lambda a + (1-\lambda)b) \leq \lambda f(a) + (1-\lambda) f(b)$$ $$f(b + \lambda(a-b)) \leq f(b) + \lambda (f(a) - f(b))$$ We know from the intermediate value theorem that there exists a ##c \in (b,a)## such that $$\frac{f(a) - f(b)}{a-b} = f'(c).$$ Hence $$f(b + \lambda(a-b)) \leq f(b) + \lambda (a - b) f'(c))$$ $$\frac{f(b + \lambda(a-b)) - f(b)}{\lambda(a-b)}...
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