HO Gaussian Integrals - Ouachita Baptist Universityplaza.obu.edu/corneliusk/mp/gi.pdfMicrosoft Word...
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Transcript of HO Gaussian Integrals - Ouachita Baptist Universityplaza.obu.edu/corneliusk/mp/gi.pdfMicrosoft Word...
Gaussian Integrals
For (a > 0)
2
12
0
(2 1)!!2n n
n ax na a
x e dx π+
∞− −
=∫ n = 0, 1, 2, 3 ….
2
0
12
ax
ae dx π∞− =∫
22 11
0
!2
nn
ax na
x e dx++
∞− =∫ n = 0, 1, 2, 3 ….
2
0
12
ax
axe dx
∞− =∫
Also
22 (2 1)!!2n n
n ax na a
x e dx π∞−
−∞
−=∫ n = 0, 1, 2, 3 ….
2axa
e dx π∞−
−∞
=∫
22 1 0n axx e dx+∞
−
−∞
=∫ (odd function over symmetric limits )
Factorial Relations The double factorial of a positive integer n is a generalization of n! and is defined as:
Other useful relations:
(2 1)!(2 1)!!2 !n
nnn+
+ =
(2 )!! 2 !nn n= n = 0, 1, 2 ….
(2 )!(2 1)!!2 !n
nnn
− =
(n = 0):
(n = 0):
(n = 0):
