Special Functions: Complete Answers?

[Mr Davis 97]

I have a relatively light question about special functions. As an example, it can be shown that $\displaystyle \int_0^{\frac{\pi}{2}} \sqrt{\sin x} ~ dx = \frac{\sqrt{\pi} ~\Gamma (\frac{3}{4})}{2 \Gamma (\frac{5}{4})}$. Generally, the expression on the right would be taken as "the answer" to this problem. My question is, to what extent is this a complete answer? Isn't the gamma function technically just another integral that we don't know the value of? And if we derive the values of gamma numerically, why don't we just numerically evaluate the original integral to begin with?

 

[fresh_42]

My question is, to what extent is this a complete answer?

The same is true for results expressed in e.g. $\log$ or $\cos$. To what extend is that a complete answer?

Isn't the gamma function technically just another integral that we don't know the value of?

And the same hols true for $\log$. Most values can only be given numerically, so why should we look for anti-derivatives at all?

This entire question depends heavily on what you will allow as fundamental function and what not. We know a lot of values of the Gamma function and many calculation rules, too. So given a result expressed in terms of the Gamma function can be used for further treatment - usually better than the original integral. In the end it always comes down to the question: What do you want to do with the result? A numerical value is certainly better for engineers, whereas the Gamma function might be better for theoretical physicists and mathematicians.