ABSTRACT
IO=~. (A.3) This result also allows us to calculate the integral of the next form:
Let us generalise these relations for the case of integration over an ndimensional volume:
(A.5)
(A.6) where OT is the transposed matrix and OiOJ = OJOi = 1. For the sake of simplicity let us denote the set of coordinates Xi via the vector X = (Xl> X2 ... xn ). Then integral (A.5) takes the form
The Jacobian of the transformation from the set of variables X to the new coordinates Y = OX = Yl> Y2 ... Yn is equal to one because OT 0 = 1. This means that the integral (A.5), in terms of variables Y, reduces to the product of n Gaussian integrals
I f d d d _yTA'Y n/2 ( }-1/2 = Y1 Y2··· Yne = 7r a1a2··· an , -00
(A.9) In some applications the Gaussian integral over the complex plane z is defined as follows:
The generalisation of this integral for n complex variables reads
(A.lO)
(A.12)
which have been applied in chapter 2 to calculate the path integral for the case of the harmonic oscillator. We start from the Fourier expansion of the function cos (ax) on an interval [-1Tj 1T]:
where the expansion coefficients are
2j'lrd () ( ) 1 (sin[(a+ n)1T] sin[(a-n)1T]) - x cos ax cos nx = - + -.!...:...------'---.!. 1T 1T a+n a-n
= -- 2 2 sm(a1T) . 1T a-n (A.14)
or
1 2a( 1 1 1 ) cot (a1T) - a1T = ---;- 1 _ a2 + 22 - a2 + 32 _ a2 + ... . (A.16)
Evaluating the integral over a from eq. (A.16) from 0 to x, we obtain for the left-hand side
and, in the same manner, for the n-th term of the right-hand side we have
o n -a 0 n (A.18)
Thus
(A.19)
An integral of the form
o
(A.20)
appeared in chapter 5 when we considered the Casimir effect (see eq. (5.33)). In order to evaluate it, note that on the interval where the integration variable x is defined, the integrand may be expanded as a series
(A.21)
Evaluating this series term-by-term gives
j1dX (1 1 1 ) 00 1 1= -In(l - x) = - 1 + "4 + -9 + 16 + . .. = - L: 2" = -«(2) , o X n=l n
(A.22) where
1 1 00 1 «(8)=1+-2 +-3 + ... =L:-
To calculate specific values of the (-function, including «(2), we reexpress the series (A.22) in terms of Bernoulli number8 Bn, which are the power expansion coefficients of some of the elementary functions. For example we can define these numbers as
Expansion of the left-hand side of this expression yields
(A.25)
or
(A.26)
Equating powers of x we obtain a chain of equations to define the Bernoulli numbers:
Bo = 1, (A.27) The first few of these numbers are
Bg = Bs = B7 = ... = 0 . (A.28) By the substitution x ---+ 2ix in eq. (A.24) and taking the real part we obtain the following power expansion of xcotx (0 < Ixl < 1T):
00 B (2x)2n xcotx = ~(-1t 2(2n)! (A.29)
The r.h.s. of this expression can be re-written as a power expansion in a which yields
2 ( 1 1 ) 1 - 2a 1 + 22 + 32 + ... (A.31)
By comparison of this formula with the power series (A.29) we obtain
(A.32)
Thus (A.33)
This is the value of the integral (A.20).
