SPECIAL FUNCTIONS

which is the required result.

We note that is it conventional to define, in addition, the functions

P(a, x)≡

γ(a, x)

Γ(a)

,Q(a, x)≡

Γ(a, x)

Γ(a)

,

which are also often called incomplete gamma functions; it is clear thatQ(a, x)=

1 −P(a, x).

18.12.4 The error function

Finally, we mention theerror function, which is encountered in probability theory

and in the solutions of some partial differential equations. The error function is

related to the incomplete gamma function by erf(x)=γ(^12 ,x^2 )/

√

πand is thus

given by

erf(x)=

2

√

π

∫x

0

e−u

2

du=1−

2

√

π

∫∞

x

e−u

2

du. (18.167)

From this definition we can easily see that

erf(0) = 0, erf(∞)=1, erf(−x)=−erf(x).

By making the substitutiony=

√

2 uin (18.167), we find

erf(x)=

√

2

π

∫√ 2 x

0

e−y

(^2) / 2
dy.
The cumulative probability function Φ(x) for the standard Gaussian distribution
(discussed in section 30.9.1) may be written in terms of the error function as
follows:
Φ(x)=
1
√
2 π
∫x
−∞
e−y
(^2) / 2
dy

1
2

• 
  

  1
  √
  2 π
  ∫x
  0
  e−y
  (^2) / 2
  dy

  
  

  1
  2

  
  


• 
  

  1
  2
  erf
  (
  x
  √
  2
  )
  .
  It is also sometimes useful to define thecomplementary error function
  erfc(x)=1−erf(x)=
  2
  √
  π
  ∫∞
  x
  e−u
  2
  du=
  Γ(^12 ,x^2 )
  √
  π

  
  

. (18.168)

18.13 Exercises

18.1 Use the explicit expressions