Lecturer: K.S. Narain , ICTP

This lesson starts with the study of the **multi-valued function** functions taken as a reference the function F(r, ?) =Sqrt(z(r,?)) where z is a **complex **variable in the **polar representation**. In fact if we start from the point z and move around a closed path that contains the origin (z=0) then the value of F at the original point change sing F (r, ?) = -F (r, ? +2?), this does not happened for any other path that does not contain the origin. The points that have these properties are called **Branch point **and** **for this function it was also shown that infinity is a **Branch point****.**

Then it were recall the definition and the properties of the **Gamma function **like for example ?(z)=(z-1)?(z-1). It fallows the introduction of the **properties** of the **Beta function** and its definition in terms of the **Gamma function. **A consequence** **of this definition is the important formula ?(z)?(1-z) = ?/Sin[?z] the importance of this formula relies in the fact that it give information about the poles, that you can use to calculate integrals using the **Residue theorem. **The last topic related with the **Gamma function **was the** ** **Stirling's formula for the Gamma function.**

The second part of the class started by recalling the **Rodrigues' formula **for** orthogonal polynomials** and the three conditions imposed to the functions involved in this formula C

_{n}(x), W(x) and S(x). The conditions are i) C

_{1}(x) is linear in x ii) S(x) is at most quadratic on x iii) W(a)S(a) = W(b)S(b) = 0 where a,b are the boundaries of x. During the present lesson this conditions are analyzed to understand witch kind of functions C

_{n}(x), W(x) and S(x). are allowed.

It is shown that if S(x) is equal to a constant ? then W(x) = Exp[-x^{2}/2?]. The polynomials generated by the **Rodrigues' formula **for the case S(x)=1 are called the **Hermite polynomials**. The second possible case is S(x) = ?(x+?) a linear function, and in the particular case S(x)=x the **Laguerre polynomials** are obtained. The last possibility is a quadratic form of S, S(x) = ?(x+?)(x+?). For S(x)=x^{2}-1 the **Jacobi polynomials** are derived but for the particular case where S(x)=x^{2}-1 and W(x)=1 then the **Legendre polynomials **are obtained. Finally some **recurrence relations **and **differential equations** that are satisfied by these polynomials were studied.

Written by Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4