Mathematical Physics Vol 1

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

253

5.6.3 Error function

Definition The following integral

x Z 0

2 √ π

e − t 2 d t

erf ( x )=

(5.173)

defines a function called the error function .

Figure 5.9: Error function.

This function can be represented as a series

∞ ∑ k = 1

x 2 k − 1 ( 2 k − 1 ) k ! .

2 √ π

( − 1 ) k + 1

erf ( x )=

(5.174)

Definition A complementary error function or erfc function is also used, defined by the relation

∞ Z x

2 √ π

e − t 2 d t

erfc ( x )= 1 − erf ( x )=

(5.175)

.

From the definitions of these functions and (5.173) and (5.174) it follows that erf ( ∞ )= 1 and erfc ( 0 )= 1 .

(5.176)

Definition Functions C( x ) andS( x ) defined by the following relations

x Z 0 x Z 0

π t 2 2

C ( x )=

cos

d t ,

(5.177)

π t 2 2

S ( x )=

sin

d t ,

are called Fresnel 13 integrals .