K.S. Narain , ICTP

This lesson starts by the introduction of **Generalized function** and its applications in physics, an example is the **Dirac delta function**, that have a direct application in physics, for example in the point particle limit the **Charge density** is a **Dirac delta function**. Some **properties** of the delta function are used. Nevertheless this delta function is not continuous and is better to work with continuous functions for this reason the **Generalized functions** were introduced to better define the delta functions. Then the **Dirac delta function** is defined as a limiting case of sequence of **Gaussian functions** of the form D_{n}(x) = Sqrt[n/pi]Exp[-nx^{2}] the integral of this function is always one independently of the value of n and in the limit n goes to infinity it converge to the **Dirac delta function** this was demonstrated in the second part of the lesson.

To define in a rigorous way the **Generalized function** the concept of **good functions** was introduced: A **good function** is a function that is differentiable any number of time and converge to zero when Abs[x] goes to infinity faster that any power of 1/ Abs[x]^{n}, an example is the function Exp[-ax^{2}] multiplied for any polynomial as well as its derivatives. The definition for a **fairly good functions** was also given: A **fairly good function** is a function that is differentiable any number of time and converge to zero when Abs[x] goes to infinity as Abs[x]^{n} for some finite value of n. If was noticed that if a function is a **good function** then its **Fourier transform** is also a **good function**.

Then a rigorous definition for the **Generalized functions** was given: Suppose a sequence of **good functions** {f_{n}(x)} if the integral of any f_{n}(x) multiplied by any **good function** g(x) converge for n goes to infinity then the sequence {f_{n}(x)} defines a **Generalized function** G(x). It was shown that any **good function** can be always written as a **Generalized function**. As examples of **Generalized functions** it were presented the **Step function **and the constant G(x)=1. Any **fairly good function** p(x) can be considered a **Generalized function** and the sequence of **good functions** that generate it is f_{n}(x)= p(x) Exp[-x^{2}/n^{2}].

In the second part of this lesson attention was given to the properties of the **Generalized functions**. It was obtained that the linear combination of generalize functions is also a generalize function. The multiplication between a **fairly good functions** and a generalize function was defined and it was shown to give as a result also a generalize function. Note that the product of two generalize functions it is not defined. The derivative of a generalize function was also defined as the sequence of the derivative of the **good functions** {df_{n}(x)/dx}.

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>