Lecturer: K.S. Narain , ICTP

This lesson starts with the recall that the **Fourier transform** of a **good function **is also a** good function **and then beings the study of the **Fourier transforms** of a **Generalized function. **It was demonstrated that if the sequence {fn(x)} defines a **Generalized function** G(x) then the sequence of **Fourier**** transforms {Fn**(x)}, where **Fn**(x) is the **Fourier transform** of the function fn(x) for all n, defines also a **Generalized function G**(x)**, **that is called the **Fourier transform **of G(x) and the **inverse** **Fourier transform **of **G**(x) give back G(x).

The particular case of the **Dirac delta function **was then** **studied in more details and some of it **Properties** demonstrated with the help of the **Generalized function **definition. The second example studied was the ** Step function **and the sequence of functions that defines it was introduced

**.**It was shown that the derivative of the

**is the**

**Step function****Dirac delta function.**Then the

**Fourier transform**of the

**Dirac delta function**?(x) was calculated and it was shown that is simply some constant, threfore the

**Fourier transform**of a constant is again the

**Dirac delta function**. Fallowing it was obtained that ?(y(x))=?

_{i}?(x-x

_{i})/(dy/dx|

_{x=x}

_{i}) where y(x

_{i})=0.

In the second part of the lesson the differential equations were studied and as an example the **differential equation** for the **Laguerre polynomials** ( x ?^{2}/?x^{2} L^{v}_{n} + (-x+v+1) ?/?x L^{v}_{n} + n L^{v}_{n}= 0) was derived. Then the definition of a** differential equation **of nth order was given as F(U(x),dU/dx,.....,d^{n}U/dx^{n})=0. The problem consists in to obtain the function U(x). To solve this problem a theorem was introduced it says that: if the function H(U(x_{0}),dU/dxI_{x=x}_{o},.....,d^{n-1}U/dx^{n-1}I_{x=x}_{o}) satisfy the** Lipschitz Condition** then d^{n}U/dx^{n}=H(U(x_{0}),dU/dxI_{x=x}_{o},.....,d^{n-1}U/dx^{n-1}I_{x=x}_{o}). This result implies that the solution of the differential equation of nth order exist. As can be seen the information of the value of the function U(x_{0}) and its derivative to order n-1 for x=x_{0} have to be known. This condition is known as the boundary condition and the solution depends which boundary condition is choused. In general the resulting **differential equation** could be very complicated but in this course we focused just in linear **differential equations**. Finally the definition for homogeneous and inhomogeneos **differential equation** was given and as an example the solution of a one order **differential equation** was obtained.

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