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 U_{1}(z) and U_{2}(z) grows exponentially for z=infinity while ?(z) does not, so the linear combination of U_{1}(z) and U_{2}(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=z^{2}/2 then it is obtained a **confluent hypergeometric differential equation** with c=1/2 and a=-?/2 so the solutions are ?(-?/2,1/2,z^{2}/2) and ?(-?/2,1/2,z^{2}/2). The solution for the **Hermite polynomials** is ?(-?/2,1/2,z^{2}/2) but the problem is that the condition for ?(-?/2,1/2,z^{2}/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 U_{1}(z) and U_{2}(z) is used and it is let as an exercise to show that for v=n the series expansions of U_{1}(z) and U_{2}(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>