This is an interesting question : << When I looked up the some information I found that there special functions are defined by these integrals themselves. What is the use of such functions? How do they help us in developing "mathematics"? >>.
Many students not yet familiar with the special functions raise the question of whether it is in the interest of giving a name to some particular integrals and using them to solve other problems. At first sight this looks like a vicious circle.
As a ready made answer to the question, let cite a passage from the paper referenced below :
We see the considerable advantage which comes from that. This avoids to taking back the
whole problem for every met case and moreover, this allows to express the result with functions which each one can recognize and can find in handbooks and in software.
"Standard" or "typical" species are duly listed, described. They were dissected, analyzed. Their properties were deeply studied. A large number of relationships between them are established, recorded. There are tables, algorithms for each of them. In brief, in a nutshell, an immense background is available. But to reach it, a key is needed, more exactly a keyword : it is necessary to know the name of the appropriate special function.
Nowadays, the art of the Mathematician in this domain is relieved by software for formal
calculation. Certainly, it does not replace the indispensable skill. Nevertheless, in many circumstances, integral calculus, differential equations, etc. the software *** will be able to bring out a formula, sometimes esoteric at first sight and including special functions. In doing it, he does not give the Solution (with a big S), but he gives an important piece of information: the name of the relevant special functions and how they act. Then, it is up to the user to decide if this closed form is enough for him, because the special functions are familiar objects for him. If not, it is up to the user to search in the books what are the properties of the special function and to find
the information which are lacking to him, or to find the intermediate developments which the use of a special function has "short-circuited".
More discussion on this subject in : https://fr.scribd.com/doc/14623310/Safari-on-the-country-of-the-Special-Functions-Safari-au-pays-des-fonctions-speciales , translation pages 18-36.