Mathematical Physics Vol 1

5.5 Some other special functions

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On the other hand

d d x

ν J

ν − 1 J

ν J ′

ν ( x ))= ν x

( x

ν ( x )+ x

ν ( x ) .

(5.108)

From here it follows

x ν J

ν x ν − 1 J

ν J ′

ν ( x )+ x

ν ( x ) ,

ν − 1 =

and then, by dividing with x ν we obtain (5.99).

5.5 Some other special functions

Almost without exception, the most commonly used special functions are trigonometric (Fourier series), hyperbolic, Bessel and Legendre functions. However, there are several notable problems in physics and engineering, the solution of which imposes the introduction of some other functions. In this chapter we will only list a group of these functions, without going into details and analyzing their properties.

5.5.1 Hermit polynomials

The function denoted by He n ( x ) , which represents the solution of the differential equation y ′′ − xy ′ + ny = 0 , (5.109) is given by the expression

n ! 2! ( n − 2 ) !

n ! 4! ( n − 4 ) !

n ! 6! ( n − 6 ) !

n −

x n − 2 + 1 · 3

x n − 4 − 1 · 3 · 5

x n − 6 + ··· (5.110)

He n ( x )= x

Functions defined in this way are called Hermit polynomials 10 . These polynomials can also be represented by the following relation He n ( x )=( − 1 ) n e x 2 / 2 d n d x n e − x 2 / 2 , n = 0 , 1 ,...

(5.111)

Figure 5.5: Hermit polynomials.

Some recurrence formulas

d d x

He n + 1 ( x )= x He n ( x ) −

He n ( x ) ,

(5.112)

d d x

He n ( x )= n He n

x ) .

− 1 (

10 Charles Hermite (1822-1901), Frencn mathematician, known for his work in algebra and number theory.