mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 22
Definitions and different representations of delta functions
Different representation of the -function.
a)  - function as a limiting form of the rectangular function
1
𝑓𝜎 (𝑥) = {2𝜎
0
for − 𝜎 < 𝑥 − 𝑎 < 𝜎
for |𝑥 − 𝑎| > 0
}
Thus we notice that as  decreases, the rectangular distribution becomes narrower
and sharper.
The integral
∫ 𝑓𝜎 (𝑥) 𝑑𝑥 =
2𝜎
∫ 𝑑𝑥 = 1
∞
−∞
1 𝑎+𝜎
𝑎−𝜎
This is true for any value of . Thus even in the limit  0 the structure becomes
infinitely peaked, however still retaining the area under the curve as unity.
So, Lim 0 𝜎𝑓(x) =  (x-a)
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2. NPTEL – Physics – Mathematical Physics - 1
Also ∫ 𝑔(𝑥)𝑓𝜎 (𝑥)𝑑𝑥 =
2𝜎
∫ 𝑔(𝑥) 𝑑𝑥
∞
−∞ 𝑎−𝜎
1 𝑎+𝜎
We assume that the function g(x) is continuous at x = a. Thus when in the infinit-
esimal interval - < x - a <, g(x) may be assumed to be a constant (g = a,(say))
and thus can be taken out of the integral. So,
∞
∫ 𝑔(𝑥) 𝛿(𝑥 − 𝑎)𝑑𝑥 =
−∞
𝐿𝑖𝑚
𝑎 → 0
∞
∫ 𝑔(𝑥)
𝑓𝜎 (𝑥)𝑑𝑥
−∞
= 𝐿𝑖𝑚
𝑎 → 0 ∫−∞ 𝑔(𝑎) ∫𝑎−𝜎 𝑑
𝑥
∞ 𝑎+
= g(a)
This property of the - function has been stated earlier. Thus the rectangular
distribution 𝑓𝜎(𝑥) in the limit  0 represents  - a function.
b) Gaussian representation of the  -function A Gaussian is denoted by,
𝑓𝜎 (x)= √2𝜋𝜎2 exp[−
Again,
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1 (𝑥−𝑎)2
2𝜎2 ]; 𝜎>0

3. NPTEL – Physics – Mathematical Physics - 1
Again as  decreases, the Gaussian becomes sharper and in the limit  0 one
will get a - function. Also the integral,
∞
∫ 𝑓
(𝑥)𝑑𝑥=1
−∞
Further it has a width  and at x = 0 it has a value
1
√2𝜋𝜎2 . So,
𝛿(x-a) =𝐿𝑖𝑚𝜎→0 √2𝜋𝜎2 exp[−
1 (𝑥−𝑎)2
2𝜎2 ]
c) Integral representation of the - function
Let's consider the integral relation,
1 ∞ sin[𝑔(𝑥 − 𝑎)]
𝜋
∫
−∞ (𝑥 − 𝑎)
𝑑𝑥 = 1 𝑔 > 0
This is true irrespective of the value of g.
Consider the relation,
𝐿𝑖𝑚
𝑥→0 𝑥
𝑠𝑖𝑛𝑔𝑥
= 𝑔
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4. NPTEL – Physics – Mathematical Physics - 1
Thus for a large value of g, the function
𝑠𝑖𝑛𝑔(𝑥−𝑎)
𝜋(𝑥−𝑎)
is a sharply peaked function at x=a.
So, 𝛿(x-a) = lim
𝑠𝑖𝑛𝑔(𝑥−𝑎)
𝑔→ (𝑥−𝑎)
Now ∫
2𝜋 𝜋(𝑥−𝑎)
1 ∞𝑔
𝑒 ± 𝑖𝑘(𝑥 − 𝑎)𝑑𝑘 =
𝑠𝑖𝑛𝑔(𝑥−𝑎)
−𝑔
Using the above two equation,
1 ∞
2𝜋
∫ ±𝑖𝑘(𝑥 − 𝑎)𝑑𝑘 = 𝛿(𝑥 − 𝑎)
−∞
This is the integral representation of the function.
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