Modular functions and modular forms

These thesis mainly concerned with modular functions and related modular forms. In Chapter L firstly homogeneous and inhomogeneous linear transformations concepts, and relations between these two transformations are considered. The modular group is defined using these transformations. After these, modular transformations are classified by means of the transformations, and also generators of the modular group and their relations are studied. Furthermore, fundamental region is defined. Tesellation upper half plane is examined because of existence of fundamental region. In Chapter II, modular functions which are not periodic and tesellate the complex plane like triangular and take all possible values in the each region are discussed. However, in this chapter, modular forms which represent an important class of functions in the development of the theory of elliptic modular functions are studied. In addition existence of modular forms are showed by means of modular functions. Connected with this, construction of entire modular forms with the help of J which is derivative of modular J function and their properties are studied. Lastly, the space of entire modular forms and relations between modular functions and entire modular forms are examined.