20
1 Gamma Functions, Beta Functions, and Related
Case 6 This last transformation can be used to compute B z (a, b) outside the unit
circle
2 F 1 ( ˜
a, ˜
b; ˜
c; z) =
Γ ( ˜
c)Γ ( ˜
b − ˜
a)
Γ ( ˜
bΓ ( ˜
c − ˜
a)
(−1)
˜
a (z)
−˜ a
2 F 1
˜
a, ˜
a + 1 − ˜
c; ˜
a + 1 − ˜
b;
1
z
(1.64)
+
Γ ( ˜
c)Γ ( ˜
a − ˜
b)
Γ ( ˜
aΓ ( ˜
c − ˜
b)
(−1)
˜
b (z)
− ˜
b
2 F 1
˜
b, ˜
b + 1 − ˜
c; ˜
b + 1 − ˜
a;
1
z
.
Due to Eq. (1.44a) we set
˜
a = a, ˜
b = 1 − b, and ˜
c = a + 1,
(1.65)
and thus we get from Eq. (1.64) for the first summand the constant term
(−1)
a Γ (a)Γ (1 − b − a)
Γ (1 − b)
(1.66)
and from the second summand
t 0 = 1, t n+1 =
(1 − a − b + n)(1 − b + n)
(2 − a − b + n)(n + 1)
t n .
(1.67)
Due to the first summand we will get a singular result for a a negative integer and
for a + b positive integers.
1.8.3 The Class incbetaC
MATLAB comes with the functions betainc for the incomplete Beta function.
Note, betainc returns the Beta normalized functions, Eq. (1.43), and is restricted
to positive real values for “a, b” and real values between 0 · · · 1 for “z”. The
SPECFUNPHYS-class incbetaC computes the (non-normalized) incomplete Beta
functions, therefore differs by the scaling factor Beta(a,b) from the MATLAB
function, but is not restricted to real values.
The SPECFUNPHYS-class [obj, erg] = incbetaC(z,a,b,nm) returns
the incomplete Beta function in the complex plane. (z,a,b) are the same variables as
in Eq. (1.44a), they could be either of the same size or scalar. “nm” is optional and
the upper bound of summations of series expansion, with default value 1000. The
object “obj” of the class comes with the properties value (computational result), info
(information about computational method), ina, inb (numerical input a, b), and inz
(numerical input z), and “erg” is the computational value in doubles.
incbetaC supports the following methods: “abs” for computing the absolute
value, “real” for the real, and “imag” for the imaginary value, “angle” for the phase
1 Gamma Functions, Beta Functions, and Related
Case 6 This last transformation can be used to compute B z (a, b) outside the unit
circle
2 F 1 ( ˜
a, ˜
b; ˜
c; z) =
Γ ( ˜
c)Γ ( ˜
b − ˜
a)
Γ ( ˜
bΓ ( ˜
c − ˜
a)
(−1)
˜
a (z)
−˜ a
2 F 1
˜
a, ˜
a + 1 − ˜
c; ˜
a + 1 − ˜
b;
1
z
(1.64)
+
Γ ( ˜
c)Γ ( ˜
a − ˜
b)
Γ ( ˜
aΓ ( ˜
c − ˜
b)
(−1)
˜
b (z)
− ˜
b
2 F 1
˜
b, ˜
b + 1 − ˜
c; ˜
b + 1 − ˜
a;
1
z
.
Due to Eq. (1.44a) we set
˜
a = a, ˜
b = 1 − b, and ˜
c = a + 1,
(1.65)
and thus we get from Eq. (1.64) for the first summand the constant term
(−1)
a Γ (a)Γ (1 − b − a)
Γ (1 − b)
(1.66)
and from the second summand
t 0 = 1, t n+1 =
(1 − a − b + n)(1 − b + n)
(2 − a − b + n)(n + 1)
t n .
(1.67)
Due to the first summand we will get a singular result for a a negative integer and
for a + b positive integers.
1.8.3 The Class incbetaC
MATLAB comes with the functions betainc for the incomplete Beta function.
Note, betainc returns the Beta normalized functions, Eq. (1.43), and is restricted
to positive real values for “a, b” and real values between 0 · · · 1 for “z”. The
SPECFUNPHYS-class incbetaC computes the (non-normalized) incomplete Beta
functions, therefore differs by the scaling factor Beta(a,b) from the MATLAB
function, but is not restricted to real values.
The SPECFUNPHYS-class [obj, erg] = incbetaC(z,a,b,nm) returns
the incomplete Beta function in the complex plane. (z,a,b) are the same variables as
in Eq. (1.44a), they could be either of the same size or scalar. “nm” is optional and
the upper bound of summations of series expansion, with default value 1000. The
object “obj” of the class comes with the properties value (computational result), info
(information about computational method), ina, inb (numerical input a, b), and inz
(numerical input z), and “erg” is the computational value in doubles.
incbetaC supports the following methods: “abs” for computing the absolute
value, “real” for the real, and “imag” for the imaginary value, “angle” for the phase