Mathematical Methods Lecture 18 of 34

November 3, 2011 by K.S. Narain

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 <span class="toctext">polar representation</span>. 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 <span class="toctext">Stirling's formula for the Gamma function.</span>

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 Cn(x), W(x) and S(x). The conditions are i) C1(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 Cn(x), W(x) and S(x). are allowed.

It is shown that if S(x) is equal to a constant ? then W(x) = Exp[-x2/2?]. 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)=x2-1 the Jacobi polynomials are derived but for the particular case where S(x)=x2-1 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.

Mathematics for Physicists (Text Book from Google e-books preview)Written by<span class="addmd"> Philippe Dennery,Andre Krzywicki, Chapter 1, 2, 3 and 4

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