MHB Solving a definite integral without using gamma function

[Saitama]

Problem:
Evaluate:
$$\int_0^{\infty} t^{-1/4}e^{-t}\,dt$$

Attempt:
I recognised this one as $\Gamma(3/4)$. I found a few formulas on Wolfram Mathworld website which helps to evaluate this but I am wondering if I can solve the definite integral from elementary methods (like by parts).

Any help is appreciated. Thanks!

 

[alyafey22]

This integral cannot be solved using elementary methods because the result is only representable using the gamma function

$$\Gamma(x) = \int^\infty_0 t^{x-1}\, e^{-t} \, dt$$

and cannot be introduced otherwise. The result can be written differently using some properties of the gamma function. For example ,

$$\Gamma(x) \Gamma(1-x) = \pi \csc(\pi x) $$

$$2^{1-2x}\sqrt{\pi}\Gamma(2x) = \Gamma\left( x+\frac{1}{2}\right) \Gamma(x) $$

$$\Gamma(1+x) = x \Gamma(x)$$

 

[Saitama]

Thanks ZaidAlyafey! :)

Is there a proof for the following formula?

$$\Gamma(x) \Gamma(1-x) = \pi \csc(\pi x) $$

I am actually dealing with the product $\Gamma(3/4)\Gamma(1/2)\Gamma(1/4)$. If I use the above formula, I can easily find the value of the product.

 

[mathbalarka]

This integral cannot be solved using elementary methods because the result is only representable using the gamma function

The proof is a pain. The one I saw used relatively large amount of differential galois applied to gamma values.

 

[alyafey22]

The proof that uses less formulas and easiest , in my opinion, is using complex analysis.

 

[Saitama]

The proof that uses less formulas and easiest , in my opinion, is using complex analysis.

Its better that I stay away from the proof. :p

Thanks, I solved the problem using the formula. :)