1.7 The Beta Function
15
%
intervals in use.
% Approximate bound on error is 9.0e-02. The integral
%
may not exist,
% or it may be difficult to approximate numerically
% to the requested accuracy.
%
% Ci = 7.197972035744612 - 6.956739662340192i
%
Sih = @(t) t.^(a-1) . * sin(t);
Si = integral(Sih,0,z)
% 0.590239179687008 + 0.664022327453170i
%
[objp,ergp] = incgammaC(i * z,a);
[objm,ergm] = incgammaC(-i * z,a);
Cig = (exp(-1/2 * pi * a * i). * ergp +
exp(1/2 * pi * a * i). * ergm)/2
% 6.885162924201410 - 6.622074106422456i
Sig = (exp(1/2 * pi * a * i). * ergm -
exp(-1/2 * pi * a * i). * ergp)/(2 * i)
% 0.590239111019731 + 0.664022265510878i
In this case the function integral did no longer sufficiently converge for the
generalized cos-integral. Thus in some situations it might be easier to compute
integrals via incomplete gamma functions than by direct integration. In addition
the relation with incomplete gamma function allows a generalization of integral
functions.
1.7
The Beta Function
1.7.1 Basics
The beta function is closely related to the gamma function. Its integral representation
is given by the Euler integral of 1st kind:
B(z, w) =
1
0
t
z−1 (1 − t)
w−1 dt with (z) > 0, (w) > 0 .
(1.38)
The beta function is symmetric
B(z, w) = B(z, w)
(1.39)
and holds
B(z, w) =
Γ (z)Γ (w)
Γ (z + w)
.
(1.40)