K.S. Narain , ICTP
This lesson starts recalling the results obtained in the previous lesson for the confluent hypergeometric differential equation in particular the last result that U1(z) and U2(z) grows exponentially for z=infinity while ?(z) does not, so the linear combination of U1(z) and U2(z) this must be in such a way that the exponential grows in z cancels. The expression for this linear combination was written and the constant of the linear combination were Gamma function of the parameters a and c.
In the fallowing some examples of functions that are related to the confluent hypergeometric differential equation were studied. An example of this are the Hermite polynomials this is shown by taken the Hermite's differential equation and doing the change of variable t=z2/2 then it is obtained a confluent hypergeometric differential equation with c=1/2 and a=-?/2 so the solutions are ?(-?/2,1/2,z2/2) and ?(-?/2,1/2,z2/2). The solution for the Hermite polynomials is ?(-?/2,1/2,z2/2) but the problem is that the condition for ?(-?/2,1/2,z2/2) to converge was that Re[a]>0 but here a=-?/2 so for v=n this condition does not hold. To solve this problem the definition of ? as a linear combination of U1(z) and U2(z) is used and it is let as an exercise to show that for v=n the series expansions of U1(z) and U2(z) have to be truncated.
Other examples related with this kind of differential equation are the Laguerre polynomials the Error function and the Bessel function. Starting from the Bessel Differential Equation and one more time doing some change of variables it is possible to convert it in a confluent hypergeometric differential equation with parameters a=v+1/2, c=2v+1 and z=2iz'. Then it is possible to write the Bessel function in terms of the confluent hypergeometric functions. The Bessel functions of the first and second kind were defined.
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</span>
