Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

244

A relation between the exponential and Hermit functions

∞ ∑ n = 0

t n n !

e tx − t 2 / 2

He n ( x )

(5.113)

=

.

Representation by integral

+ ∞ Z

i = √

1 √ 2 π

n e − t 2 / 2 d t ,

He n ( x )=

( x + it )

− 1 .

(5.114)

− ∞

Note that in the literature the equation

y ′′ − 2 xy ′ + 2 ny = 0 is often called Hermit differential equation, with the solution given by H ∗ n ( x )=( − 1 ) n e x 2 d n d x n e − x 2 , n = 0 , 1 ,...

5.5.2 Laguerre polynomials

the solution of the differential equation

xy ′′ +( α + 1 − x ) y ′ + ny = 0

(5.115)

is the function of the form

e x x − α n !

d n d x n

e − x x n + α

Ln ( α )

( x )=

n = 0 , 1 ,...,

(5.116)

,

which is called the Laguerre polynomial 11 (function).

5.6 Special functions that are not a result of the Frobenius method

In addition to the previously introduced special functions, which appeared as a result of solving differential equations, we will mention additional important functions that appear in solving some problems of mathematics and physics.

5.6.1 Gamma function (factorial function)

Definition Γ function is defined by the following relation

∞ Z 0

e − t t z − 1 d t , z ∈ C ,

Γ ( z )=

(5.117)

where t z − 1 = e ( z − 1 ) ln t .

11 Edmond Laguerre (1834-1886), French mathematician, known for his work in geometry and infinite series theory.