Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

254

In some problems of physics, the function of the following form appears

erf

x √ π = C ( x )+ i S ( x ) , i = √

1 1 + i

1 + i 2

− 1 .

(5.178)

whereC( x ) andS( x ) are Fresnel integrals.

Figure 5.10: Fresnel integrals.

5.6.4 Exponential integrals

The integral given by the relation

∞ Z x

e − t t

− Ei ( − x )=

d t

(5.179)

defines the so called exponential integral . This function also appears in many problems of physics. For small values of x this integral can be approximated by the relation − Ei ( x ) ≈− γ − ln x , (5.180) where γ is a constant, given by the relation (5.120). If x is substituted by iy , the exponential integral can be represented in the following form

Ei ( iy )= Ci ( y )+ i Si ( y )+ i π 2 ,

(5.181)

where two new functions, Ci ( y ) and Si ( y ) , have been introduced, defined by the following expressions Ci ( y )= − ∞ Z y cos t t d t = γ + ln y − y Z 0 1 − cos t t d t ,

(5.182)

∞ Z y

y Z 0

π 2 −

sin t t

1 − sin t t

Si ( y )=

d t =

d t .

These functions are called Ci ( y ) – cosine integral andSi ( y ) – sine integral . The constant γ is Euler’s constant. 13 Augustin Fresnel (1788-1827), French physicist, known for his work in optics.