15 Special Functions
Next: Elliptic Functions, Previous: Polynomials [Contents][Index]
17 Special Functions
Introduction to Special Functions
Bessel Functions
Airy Functions
Gamma and Factorial Functions
Exponential Integrals
Error Function
Struve Functions
Hypergeometric Functions
Parabolic Cylinder Functions
Functions and Variables for Special Functions
Next: Bessel Functions, Up: Special Functions [Contents][Index]
17.1 Introduction to Special Functions
Special function notation follows:
bessel_j (index, expr) Bessel function, 1st kind
bessel_y (index, expr) Bessel function, 2nd kind
bessel_i (index, expr) Modified Bessel function, 1st kind
bessel_k (index, expr) Modified Bessel function, 2nd kind
hankel_1 (v,z) Hankel function of the 1st kind
hankel_2 (v,z) Hankel function of the 2nd kind
struve_h (v,z) Struve H function
struve_l (v,z) Struve L function
assoc_legendre_p[v,u] (z) Legendre function of degree v and order u
assoc_legendre_q[v,u] (z) Legendre function, 2nd kind
%f[p,q] ([], [], expr) Generalized Hypergeometric function
gamma (z) Gamma function
gamma_incomplete_lower (a,z) Lower incomplete gamma function
gamma_incomplete (a,z) Tail of incomplete gamma function
hypergeometric (l1, l2, z) Hypergeometric function
%s[u,v] (z) Lommel "small" s function
slommel[u,v] (z) Lommel "big" S function
%m[u,k] (z) Whittaker function, 1st kind
%w[u,k] (z) Whittaker function, 2nd kind
erfc (z) Complement of the erf function
expintegral_e (v,z) Exponential integral E
expintegral_e1 (z) Exponential integral E1
expintegral_ei (z) Exponential integral Ei
expintegral_li (z) Logarithmic integral Li
expintegral_si (z) Exponential integral Si
expintegral_ci (z) Exponential integral Ci
expintegral_shi (z) Exponential integral Shi
expintegral_chi (z) Exponential integral Chi
kelliptic (z) Complete elliptic integral of the first
kind (K)
parabolic_cylinder_d (v,z) Parabolic cylinder D function
Categories: Bessel functions · Airy functions · Special functions ·
Next: Airy Functions, Previous: Introduction to Special Functions, Up: Special Functions [Contents][Index]
17.2 Bessel Functions
Function: bessel_j (v, z)
The Bessel function of the first kind of order v and argument z. See A&S eqn 9.1.10 and DLMF 10.2.E2.
bessel_j is defined as
although the infinite series is not used for computations.
When besselexpand is true, bessel_j is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.
Categories: Bessel functions · Special functions ·
Function: bessel_y (v, z)
The Bessel function of the second kind of order v and argument z. See A&S eqn 9.1.2 and DLMF 10.2.E3.
bessel_y is defined as
when v is not an integer. When v is an integer n, the limit as v approaches n is taken.
When besselexpand is true, bessel_y is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.
Categories: Bessel functions · Special functions ·
Function: bessel_i (v, z)
The modified Bessel function of the first kind of order v and argument z. See A&S eqn 9.6.10 and DLMF 10.25.E2.
bessel_i is defined as
although the infinite series is not used for computations.
When besselexpand is true, bessel_i is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.
Categories: Bessel functions · Special functions ·
Function: bessel_k (v, z)
The modified Bessel function of the second kind of order v and argument z. See A&S eqn 9.6.2 and DLMF 10.27.E4.
bessel_k is defined as
when v is not an integer. If v is an integer n, then the limit as v approaches n is taken.
When besselexpand is true, bessel_k is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.
Categories: Bessel functions · Special functions ·
Function: hankel_1 (v, z)
The Hankel function of the first kind of order v and argument z. See A&S eqn 9.1.3 and DLMF 10.4.E3.
hankel_1 is defined as
Maxima evaluates hankel_1 numerically for a complex order v and complex argument z in float precision. The numerical evaluation in bigfloat precision is not supported.
When besselexpand is true, hankel_1 is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.
Maxima knows the derivative of hankel_1 wrt the argument z.
Examples:
Numerical evaluation:
(%i1) hankel_1(1,0.5);
(%o1) 0.24226845767487384 - 1.4714723926702433 %i
(%i2) hankel_1(1,0.5+%i);
(%o2) - 0.2558287994862166 %i - 0.23957560188301597
Expansion of hankel_1 when besselexpand is true:
(%i1) hankel_1(1/2,z),besselexpand:true;
sqrt(2) sin(z) - sqrt(2) %i cos(z)
(%o1) ----------------------------------
sqrt(%pi) sqrt(z)
Derivative of hankel_1 wrt the argument z. The derivative wrt the order v is not supported. Maxima returns a noun form:
(%i1) diff(hankel_1(v,z),z);
hankel_1(v - 1, z) - hankel_1(v + 1, z)
(%o1) ---------------------------------------
2
(%i2) diff(hankel_1(v,z),v);
d
(%o2) -- (hankel_1(v, z))
dv
Categories: Bessel functions · Special functions ·
Function: hankel_2 (v, z)
The Hankel function of the second kind of order v and argument z. See A&S eqn 9.1.4 and DLMF 10.4.E3.
hankel_2 is defined as
Maxima evaluates hankel_2 numerically for a complex order v and complex argument z in float precision. The numerical evaluation in bigfloat precision is not supported.
When besselexpand is true, hankel_2 is expanded in terms of elementary functions when the order v is half of an odd integer. See besselexpand.
Maxima knows the derivative of hankel_2 wrt the argument z.
For examples see hankel_1.
Categories: Bessel functions · Special functions ·
Option variable: besselexpand
Default value: false
Controls expansion of the Bessel, Hankel and Struve functions when the order is half of an odd integer. In this case, the functions can be expanded in terms of other elementary functions. When besselexpand is true, the Bessel function is expanded.
(%i1) besselexpand: false$
(%i2) bessel_j (3/2, z);
3
(%o2) bessel_j(-, z)
2
(%i3) besselexpand: true$
(%i4) bessel_j (3/2, z);
sin(z) cos(z)
sqrt(2) sqrt(z) (------ - ------)
2 z
z
(%o4) ---------------------------------
sqrt(%pi)
(%i5) bessel_y(3/2,z);
sin(z) cos(z)
sqrt(2) sqrt(z) (- ------ - ------)
z 2
z
(%o5) -----------------------------------
sqrt(%pi)
(%i6) bessel_i(3/2,z);
cosh(z) sinh(z)
sqrt(2) sqrt(z) (------- - -------)
z 2
z
(%o6) -----------------------------------
sqrt(%pi)
(%i7) bessel_k(3/2,z);
1 - z
sqrt(%pi) (- + 1) %e
z
(%o7) -----------------------
sqrt(2) sqrt(z)
Categories: Bessel functions · Simplification flags and variables · Special functions ·
Function: scaled_bessel_i (v, z)
The scaled modified Bessel function of the first kind of order v and argument z. That is,
This function is particularly useful for calculating
for large z, which is large. However, maxima does not otherwise know much about this function. For symbolic work, it is probably preferable to work with the expression exp(-abs(z))*bessel_i(v, z).
Categories: Bessel functions ·
Function: scaled_bessel_i0 (z)
Identical to scaled_bessel_i(0,z).
Categories: Bessel functions · Special functions ·
Function: scaled_bessel_i1 (z)
Identical to scaled_bessel_i(1,z).
Categories: Bessel functions · Special functions ·
Function: %s [u,v] (z)
Lommel’s little
function. (DLMF 11.9.E3)(G&R 8.570.1).
This Lommel function is the particular solution of the inhomogeneous Bessel differential equation:
This can be defined by the series
where
Categories: Bessel functions · Special functions ·
Function: slommel [u,v] (z)
Lommel’s big
function. (DLMF 11.9.E5)(G&R 8.570.2).
Lommels big S function is another particular solution of the inhomogeneous Bessel differential equation (see %s) defined for all values of
and where
When
is an odd negative integer, the limit must be used.
Categories: Bessel functions · Special functions ·
Next: Gamma and Factorial Functions, Previous: Bessel Functions, Up: Special Functions [Contents][Index]
17.3 Airy Functions
The Airy functions
and are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Section 10.4 and DLMF 9.
The Airy differential equation is:
The numerically satisfactory pair of solutions (DLMF 9.2#T1) on the real line are
and These two solutions are oscillatory for x < 0. is the solution subject to the condition that as and is the second solution with the same amplitude as as which differs in phase by Also, is unbounded as
If the argument x is a real or complex floating point number, the numerical value of the function is returned.
Function: airy_ai (x)
The Airy function
See A&S eqn 10.4.2 and DLMF 9.
See also airy_bi, airy_dai, and airy_dbi.
Categories: Airy functions · Special functions ·
Function: airy_dai (x)
The derivative of the Airy function
See airy_ai.
Categories: Airy functions · Special functions ·
Function: airy_bi (x)
The Airy function
See A&S eqn 10.4.3 and DLMF 9.
See airy_ai, and airy_dbi.
Categories: Airy functions · Special functions ·
Function: airy_dbi (x)
The derivative of the Airy function
See airy_ai, and airy_bi.
Categories: Airy functions · Special functions ·
Next: Exponential Integrals, Previous: Airy Functions, Up: Special Functions [Contents][Index]
17.4 Gamma and Factorial Functions
The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.
Function: bffac (expr, n)
Bigfloat version of the factorial (shifted gamma) function. The second argument is how many digits to retain and return, it’s a good idea to request a couple of extra.
(%i1) bffac(1/2,16);
(%o1) 8.862269254527584b-1
(%i2) (1/2)!,numer;
(%o2) 0.886226925452758
(%i3) bffac(1/2,32);
(%o3) 8.862269254527580136490837416707b-1
Categories: Gamma and factorial functions · Numerical evaluation ·
Function: bfpsi (n, z, fpprec)
Function: bfpsi0 (z, fpprec)
bfpsi is the polygamma function of real argument z and integer order n. See psi for further information. bfpsi0 is the digamma function. bfpsi0(z, fpprec) is equivalent to bfpsi(0, z, fpprec).
These functions return bigfloat values. fpprec is the bigfloat precision of the return value.
(%i1) bfpsi0(1/3, 15);
(%o1) - 3.13203378002081b0
(%i2) bfpsi0(1/3, 32);
(%o2) - 3.1320337800208063229964190742873b0
(%i3) bfpsi(0,1/3,32);
(%o3) - 3.1320337800208063229964190742873b0
(%i4) psi[0](1/3);
3 log(3) %pi
(%o4) - -------- - --------- - %gamma
2 2 sqrt(3)
(%i5) float(%);
(%o5) - 3.132033780020806
Categories: Gamma and factorial functions · Numerical evaluation ·
Function: cbffac (z, fpprec)
Complex bigfloat factorial.
load ("bffac") loads this function.
(%i1) cbffac(1+%i,16);
(%o1) 3.430658398165453b-1 %i + 6.529654964201666b-1
(%i2) (1+%i)!,numer;
(%o2) 0.3430658398165453 %i + 0.6529654964201667
Categories: Gamma and factorial functions · Complex variables · Numerical evaluation ·
Function: gamma (z)
The basic definition of the gamma function (DLMF 5.2.E1 and A&S eqn 6.1.1) is
Maxima simplifies gamma for positive integer and positive and negative rational numbers. For half integral values the result is a rational number times
The simplification for integer values is controlled by factlim. For integers greater than factlim the numerical result of the factorial function, which is used to calculate gamma, will overflow. The simplification for rational numbers is controlled by gammalim to avoid internal overflow. See factlim and gammalim.
For negative integers gamma is not defined.
Maxima can evaluate gamma numerically for real and complex values in float and bigfloat precision.
gamma has mirror symmetry.
When gamma_expand is true, Maxima expands gamma for arguments z+n and z-n where n is an integer.
Maxima knows the derivative of gamma.
Examples:
Simplification for integer, half integral, and rational numbers:
(%i1) map('gamma,[1,2,3,4,5,6,7,8,9]);
(%o1) [1, 1, 2, 6, 24, 120, 720, 5040, 40320]
(%i2) map('gamma,[1/2,3/2,5/2,7/2]);
sqrt(%pi) 3 sqrt(%pi) 15 sqrt(%pi)
(%o2) [sqrt(%pi), ---------, -----------, ------------]
2 4 8
(%i3) map('gamma,[2/3,5/3,7/3]);
2 1
2 gamma(-) 4 gamma(-)
2 3 3
(%o3) [gamma(-), ----------, ----------]
3 3 9
Numerical evaluation for real and complex values:
(%i1) map('gamma,[2.5,2.5b0]);
(%o1) [1.329340388179137, 1.329340388179137b0]
(%i2) map('gamma,[1.0+%i,1.0b0+%i]);
(%o2) [0.49801566811835596 - 0.15494982830181073 %i,
4.980156681183561b-1 - 1.549498283018107b-1 %i]
gamma has mirror symmetry:
(%i1) declare(z,complex)$
(%i2) conjugate(gamma(z));
(%o2) gamma(conjugate(z))
Maxima expands gamma(z+n) and gamma(z-n), when gamma_expand is true:
(%i1) gamma_expand:true$
(%i2) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
gamma(z)
(%o2) [z gamma(z), --------, z + 1]
z - 1
The derivative of gamma:
(%i1) diff(gamma(z),z);
(%o1) psi (z) gamma(z)
0
See also makegamma.
The Euler-Mascheroni constant is %gamma.
Categories: Gamma and factorial functions · Special functions ·
Function: log_gamma (z)
The natural logarithm of the gamma function.
(%i1) gamma(6);
(%o1) 120
(%i2) log_gamma(6);
(%o2) log(120)
(%i3) log_gamma(0.5);
(%o3) 0.5723649429247004
Categories: Gamma and factorial functions · Special functions ·
Function: gamma_incomplete_lower (a, z)
The lower incomplete gamma function (DLMF 8.2.E1 and A&S eqn 6.5.2):
See also gamma_incomplete (upper incomplete gamma function).
Categories: Gamma and factorial functions · Special functions ·
Function: gamma_incomplete (a, z)
The incomplete upper gamma function (DLMF 8.2.E2 and A&S eqn 6.5.3):
See also gamma_expand for controlling how gamma_incomplete is expressed in terms of elementary functions and erfc.
Also see the related functions gamma_incomplete_regularized and gamma_incomplete_generalized.
Categories: Gamma and factorial functions · Special functions ·
Function: gamma_incomplete_regularized (a, z)
The regularized incomplete upper gamma function (DLMF 8.2.E4):
See also gamma_expand for controlling how gamma_incomplete is expressed in terms of elementary functions and erfc.
Also see gamma_incomplete.
Categories: Gamma and factorial functions · Special functions ·
Function: gamma_incomplete_generalized (a, z1, z1)
The generalized incomplete gamma function.
Also see gamma_incomplete and gamma_incomplete_regularized.
Categories: Gamma and factorial functions · Special functions ·
Option variable: gamma_expand
Default value: false
gamma_expand controls expansion of gamma_incomplete. When gamma_expand is true, gamma_incomplete(v,z) is expanded in terms of z, exp(z), and gamma_incomplete or erfc when possible.
(%i1) gamma_incomplete(2,z);
(%o1) gamma_incomplete(2, z)
(%i2) gamma_expand:true;
(%o2) true
(%i3) gamma_incomplete(2,z);
- z
(%o3) (z + 1) %e
(%i4) gamma_incomplete(3/2,z);
- z sqrt(%pi) erfc(sqrt(z))
(%o4) sqrt(z) %e + -----------------------
2
(%i5) gamma_incomplete(4/3,z);
1
gamma_incomplete(-, z)
1/3 - z 3
(%o5) z %e + ----------------------
3
(%i6) gamma_incomplete(a+2,z);
a - z
(%o6) z (z + a + 1) %e + a (a + 1) gamma_incomplete(a, z)
(%i7) gamma_incomplete(a-2, z);
gamma_incomplete(a, z) a - 2 z 1
(%o7) ---------------------- - z (--------------- + -----)
(1 - a) (2 - a) (a - 2) (a - 1) a - 2
- z
%e
Categories: Gamma and factorial functions · Simplification flags and variables ·
Option variable: gammalim
Default value: 10000
gammalim controls simplification of the gamma function for integral and rational number arguments. If the absolute value of the argument is not greater than gammalim, then simplification will occur. Note that the factlim switch controls simplification of the result of gamma of an integer argument as well.
Categories: Gamma and factorial functions · Simplification flags and variables ·
Function: makegamma (expr)
Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.
See also makefact.
(%i1) makegamma(binomial(n,k));
gamma(n + 1)
(%o1) -----------------------------
gamma(k + 1) gamma(n - k + 1)
(%i2) makegamma(x!);
(%o2) gamma(x + 1)
(%i3) makegamma(beta(a,b));
gamma(a) gamma(b)
(%o3) -----------------
gamma(b + a)
Categories: Gamma and factorial functions ·
Function: beta (a, b)
The beta function is defined as
(DLMF 5.12.E1 and A&S eqn 6.2.1).
Maxima simplifies the beta function for positive integers and rational numbers, which sum to an integer. When beta_args_sum_to_integer is true, Maxima simplifies also general expressions which sum to an integer.
For a or b equal to zero the beta function is not defined.
In general the beta function is not defined for negative integers as an argument. The exception is for a=-n, n a positive integer and b a positive integer with b<=n, it is possible to define an analytic continuation. Maxima gives for this case a result.
When beta_expand is true, expressions like beta(a+n,b) and beta(a-n,b) or beta(a,b+n) and beta(a,b-n) with n an integer are simplified.
Maxima can evaluate the beta function for real and complex values in float and bigfloat precision. For numerical evaluation Maxima uses log_gamma:
- log_gamma(b + a) + log_gamma(b) + log_gamma(a)
%e
Maxima knows that the beta function is symmetric and has mirror symmetry.
Maxima knows the derivatives of the beta function with respect to a or b.
To express the beta function as a ratio of gamma functions see makegamma.
Examples:
Simplification, when one of the arguments is an integer:
(%i1) [beta(2,3),beta(2,1/3),beta(2,a)];
1 9 1
(%o1) [--, -, ---------]
12 4 a (a + 1)
Simplification for two rational numbers as arguments which sum to an integer:
(%i1) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
3 %pi 2 %pi
(%o1) [-----, -------, sqrt(2) %pi]
8 sqrt(3)
When setting beta_args_sum_to_integer to true more general expression are simplified, when the sum of the arguments is an integer:
(%i1) beta_args_sum_to_integer:true$
(%i2) beta(a+1,-a+2);
%pi (a - 1) a
(%o2) ------------------
2 sin(%pi (2 - a))
The possible results, when one of the arguments is a negative integer:
(%i1) [beta(-3,1),beta(-3,2),beta(-3,3)];
1 1 1
(%o1) [- -, -, - -]
3 6 3
beta(a+n,b) or beta(a-n,b) with n an integer simplifies when beta_expand is true:
(%i1) beta_expand:true$
(%i2) [beta(a+1,b),beta(a-1,b),beta(a+1,b)/beta(a,b+1)];
a beta(a, b) beta(a, b) (b + a - 1) a
(%o2) [------------, ----------------------, -]
b + a a - 1 b
Beta is not defined, when one of the arguments is zero:
beta: expected nonzero arguments; found 0, b – an error. To debug this try: debugmode(true);
(%i1) beta(0,b);
Numerical evaluation for real and complex arguments in float or bigfloat precision:
(%i1) beta(2.5,2.3);
(%o1) 0.08694748611299981
(%i2) beta(2.5,1.4+%i);
(%o2) 0.06401449507966957 - 0.15020780532864159 %i
(%i3) beta(2.5b0,2.3b0);
(%o3) 8.694748611299965b-2
(%i4) beta(2.5b0,1.4b0+%i);
(%o4) 6.401449507966939b-2 - 1.502078053286414b-1 %i
Beta is symmetric and has mirror symmetry:
(%i1) beta(a,b)-beta(b,a);
(%o1) 0
(%i2) declare(a,complex,b,complex)$
(%i3) conjugate(beta(a,b));
(%o3) beta(conjugate(a), conjugate(b))
The derivative of the beta function wrt a:
(%i1) diff(beta(a,b),a);
(%o1) - beta(a, b) (psi (b + a) - psi (a))
0 0
Categories: Gamma and factorial functions ·
Function: beta_incomplete (a, b, z)
The basic definition of the incomplete beta function (DLMF 8.17.E1 and A&S eqn 6.6.1) is
This definition is possible for
and and For other values the incomplete beta function can be defined through a generalized hypergeometric function:
gamma(a) hypergeometric_generalized([a, 1 - b], [a + 1], z) z
(See https://functions.wolfram.com/GammaBetaErf/Beta3/ for a complete definition of the incomplete beta function.)
For negative integers a = -n and positive integers b=m with
the incomplete beta function is defined through
Maxima uses this definition to simplify beta_incomplete for a a negative integer.
For a a positive integer, beta_incomplete simplifies for any argument b and z and for b a positive integer for any argument a and z, with the exception of a a negative integer.
For z=0 and
beta_incomplete has the specific value zero. For z=1 and beta_incomplete simplifies to the beta function beta(a,b).
Maxima evaluates beta_incomplete numerically for real and complex values in float or bigfloat precision. For the numerical evaluation an expansion of the incomplete beta function in continued fractions is used.
When the option variable beta_expand is true, Maxima expands expressions like beta_incomplete(a+n,b,z) and beta_incomplete(a-n,b,z) where n is a positive integer.
Maxima knows the derivatives of beta_incomplete with respect to the variables a, b and z and the integral with respect to the variable z.
Examples:
Simplification for a a positive integer:
(%i1) beta_incomplete(2,b,z);
b
1 - (1 - z) (b z + 1)
(%o1) ----------------------
b (b + 1)
Simplification for b a positive integer:
(%i1) beta_incomplete(a,2,z);
a
(a (1 - z) + 1) z
(%o1) ------------------
a (a + 1)
Simplification for a and b a positive integer:
(%i1) beta_incomplete(3,2,z);
3
(3 (1 - z) + 1) z
(%o1) ------------------
12
a is a negative integer and b<=(-a), Maxima simplifies:
(%i1) beta_incomplete(-3,1,z);
1
(%o1) - ----
3
3 z
For the specific values z=0 and z=1, Maxima simplifies:
(%i1) assume(a>0,b>0)$
(%i2) beta_incomplete(a,b,0);
(%o2) 0
(%i3) beta_incomplete(a,b,1);
(%o3) beta(a, b)
Numerical evaluation in float or bigfloat precision:
(%i1) beta_incomplete(0.25,0.50,0.9);
(%o1) 4.594959440269333
(%i2) fpprec:25$
(%i3) beta_incomplete(0.25,0.50,0.9b0);
(%o3) 4.594959440269324086971216b0
For abs(z)>1 beta_incomplete returns a complex result:
(%i1) beta_incomplete(0.25,0.50,1.7);
(%o1) 5.244115108584249 - 1.4551804778784403 %i
Results for more general complex arguments:
(%i1) beta_incomplete(0.25+%i,1.0+%i,1.7+%i);
(%o1) 2.7269606756625384 - 0.38311757042691896 %i
(%i2) beta_incomplete(1/2,5/4*%i,2.8+%i);
(%o2) 13.046496351687155 %i - 5.8020679562699975
Expansion, when beta_expand is true:
(%i1) beta_incomplete(a+1,b,z),beta_expand:true;
b a
a beta_incomplete(a, b, z) (1 - z) z
(%o1) -------------------------- - -----------
b + a b + a
(%i2) beta_incomplete(a-1,b,z),beta_expand:true;
b a - 1
beta_incomplete(a, b, z) (- b - a + 1) (1 - z) z
(%o2) -------------------------------------- - ---------------
1 - a 1 - a
Derivative and integral for beta_incomplete:
(%i1) diff(beta_incomplete(a, b, z), z);
b - 1 a - 1
(%o1) (1 - z) z
(%i2) integrate(beta_incomplete(a, b, z), z);
(%o2) beta_incomplete(a, b, z) z - beta_incomplete(a + 1, b, z)
(%i3) factor(diff(%, z));
(%o3) beta_incomplete(a, b, z)
Categories: Gamma and factorial functions ·
Function: beta_incomplete_regularized (a, b, z)
The regularized incomplete beta function (DLMF 8.17.E2 and A&S eqn 6.6.2), defined as
As for beta_incomplete this definition is not complete. See https://functions.wolfram.com/GammaBetaErf/BetaRegularized/ for a complete definition of beta_incomplete_regularized.
beta_incomplete_regularized simplifies a or b a positive integer.
For z=0 and
beta_incomplete_regularized has the specific value 0. For z=1 and beta_incomplete_regularized simplifies to 1.
Maxima can evaluate beta_incomplete_regularized for real and complex arguments in float and bigfloat precision.
When beta_expand is true, Maxima expands beta_incomplete_regularized for arguments a+n or a-n, where n is an integer.
Maxima knows the derivatives of beta_incomplete_regularized with respect to the variables a, b, and z and the integral with respect to the variable z.
Examples:
Simplification for a or b a positive integer:
(%i1) beta_incomplete_regularized(2,b,z);
b
(%o1) 1 - (1 - z) (b z + 1)
(%i2) beta_incomplete_regularized(a,2,z);
a
(%o2) (a (1 - z) + 1) z
(%i3) beta_incomplete_regularized(3,2,z);
3
(%o3) (3 (1 - z) + 1) z
For the specific values z=0 and z=1, Maxima simplifies:
(%i1) assume(a>0,b>0)$
(%i2) beta_incomplete_regularized(a,b,0);
(%o2) 0
(%i3) beta_incomplete_regularized(a,b,1);
(%o3) 1
Numerical evaluation for real and complex arguments in float and bigfloat precision:
(%i1) beta_incomplete_regularized(0.12,0.43,0.9);
(%o1) 0.9114011367359802
(%i2) fpprec:32$
(%i3) beta_incomplete_regularized(0.12,0.43,0.9b0);
(%o3) 9.1140113673598029169207248506439b-1
(%i4) beta_incomplete_regularized(1+%i,3/3,1.5*%i);
(%o4) 0.2865367499935405 %i - 0.12299596333468409
(%i5) fpprec:20$
(%i6) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i);
(%o6) 2.8653674999354031589b-1 %i - 1.2299596333468401976b-1
Expansion, when beta_expand is true:
(%i1) beta_incomplete_regularized(a+1,b,z);
(%o1) beta_incomplete_regularized(a + 1, b, z)
(%i2) beta_incomplete_regularized(a-1,b,z);
(%o2) beta_incomplete_regularized(a - 1, b, z)
The derivative and the integral wrt z:
(%i1) diff(beta_incomplete_regularized(a,b,z),z);
b - 1 a - 1
(1 - z) z
(%o1) -------------------
beta(a, b)
(%i2) integrate(beta_incomplete_regularized(a,b,z),z);
(%o2) beta_incomplete_regularized(a, b, z) z
a beta_incomplete_regularized(a + 1, b, z)
- ------------------------------------------
b + a
Categories: Gamma and factorial functions ·
Function: beta_incomplete_generalized (a, b, z1, z2)
The basic definition of the generalized incomplete beta function is
Maxima simplifies beta_incomplete_regularized for a and b a positive integer.
For
and or Maxima simplifies beta_incomplete_generalized to beta_incomplete. For and or Maxima simplifies to an expression with beta and beta_incomplete.
Maxima evaluates beta_incomplete_regularized for real and complex values in float and bigfloat precision.
When beta_expand is true, Maxima expands beta_incomplete_generalized for a+n and a-n, n a positive integer.
Maxima knows the derivative of beta_incomplete_generalized with respect to the variables a, b, z1, and z2 and the integrals with respect to the variables z1 and z2.
Examples:
Maxima simplifies beta_incomplete_generalized for a and b a positive integer:
(%i1) beta_incomplete_generalized(2,b,z1,z2);
b b
(1 - z1) (b z1 + 1) - (1 - z2) (b z2 + 1)
(%o1) -------------------------------------------
b (b + 1)
(%i2) beta_incomplete_generalized(a,2,z1,z2);
a a
(a (1 - z2) + 1) z2 - (a (1 - z1) + 1) z1
(%o2) -------------------------------------------
a (a + 1)
(%i3) beta_incomplete_generalized(3,2,z1,z2);
2 2 2 2
(1 - z1) (3 z1 + 2 z1 + 1) - (1 - z2) (3 z2 + 2 z2 + 1)
(%o3) -----------------------------------------------------------
12
Simplification for specific values z1=0, z2=0, z1=1, or z2=1:
(%i1) assume(a > 0, b > 0)$
(%i2) beta_incomplete_generalized(a,b,z1,0);
(%o2) - beta_incomplete(a, b, z1)
(%i3) beta_incomplete_generalized(a,b,0,z2);
(%o3) - beta_incomplete(a, b, z2)
(%i4) beta_incomplete_generalized(a,b,z1,1);
(%o4) beta(a, b) - beta_incomplete(a, b, z1)
(%i5) beta_incomplete_generalized(a,b,1,z2);
(%o5) beta_incomplete(a, b, z2) - beta(a, b)
Numerical evaluation for real arguments in float or bigfloat precision:
(%i1) beta_incomplete_generalized(1/2,3/2,0.25,0.31);
(%o1) 0.09638178086368676
(%i2) fpprec:32$
(%i3) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0);
(%o3) 9.6381780863686935309170054689964b-2
Numerical evaluation for complex arguments in float or bigfloat precision:
(%i1) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31);
(%o1) - 0.09625463003205387 %i - 0.0033238477353540463
(%i2) fpprec:20$
(%i3) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0);
(%o3) - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3
Expansion for a+n or a-n, n a positive integer, when beta_expand is true:
(%i1) beta_expand:true$
(%i2) beta_incomplete_generalized(a+1,b,z1,z2);
b a b a
(1 - z1) z1 - (1 - z2) z2
(%o2) -----------------------------
b + a
a beta_incomplete_generalized(a, b, z1, z2)
+ -------------------------------------------
b + a
(%i3) beta_incomplete_generalized(a-1,b,z1,z2);
beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
(%o3) -------------------------------------------------------
1 - a
b a - 1 b a - 1
(1 - z2) z2 - (1 - z1) z1
- -------------------------------------
1 - a
Derivative wrt the variable z1 and integrals wrt z1 and z2:
(%i1) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
b - 1 a - 1
(%o1) - (1 - z1) z1
(%i2) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
(%o2) beta_incomplete_generalized(a, b, z1, z2) z1
+ beta_incomplete(a + 1, b, z1)
(%i3) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
(%o3) beta_incomplete_generalized(a, b, z1, z2) z2
- beta_incomplete(a + 1, b, z2)
Categories: Gamma and factorial functions ·
Option variable: beta_expand
Default value: false
When beta_expand is true, beta(a,b) and related functions are expanded for arguments like a+n or a-n, where n is an integer.
See beta for examples.
Categories: Gamma and factorial functions · Simplification flags and variables ·
Option variable: beta_args_sum_to_integer
Default value: false
When beta_args_sum_to_integer is true, Maxima simplifies beta(a,b), when the arguments a and b sum to an integer.
See beta for examples.
Categories: Gamma and factorial functions · Simplification flags and variables ·
Function: psi [n](x)
psi[n](x) is the polygamma function (DLMF 5.2E2, DLMF 5.15, A&S eqn 6.3.1 and A&S eqn 6.4.1) defined by
Thus, psi[0](x) is the first derivative, psi[1](x) is the second derivative, etc.
Maxima can compute some exact values for rational args as well for float and bfloat args. Several variables control what range of rational args
will return an exact value, if possible. See maxpsiposint, maxpsinegint, maxpsifracnum, and maxpsifracdenom. That is, x must lie between maxpsinegint and maxpsiposint. If the absolute value of the fractional part of x is rational and has a numerator less than maxpsifracnum and has a denominator less than maxpsifracdenom, will return an exact value.
The function bfpsi in the bffac package can compute numerical values.
(%i1) psi[0](.25);
(%o1) - 4.227453533376265
(%i2) psi[0](1/4);
%pi
(%o2) - 3 log(2) - --- - %gamma
2
(%i3) float(%);
(%o3) - 4.227453533376265
(%i4) psi[2](0.75);
(%o4) - 5.3026332163376395
(%i5) psi[2](3/4);
1 3
(%o5) psi (-) + 4 %pi
2 4
(%i6) float(%);
(%o6) - 5.3026332163376395
Categories: Gamma and factorial functions ·
Option variable: maxpsiposint
Default value: 20
maxpsiposint is the largest positive integer value for which
gives an exact value for rational x.
(%i1) psi[0](20);
275295799
(%o1) --------- - %gamma
77597520
(%i2) psi[0](21);
(%o2) psi (21)
0
(%i3) psi[2](20);
1683118856778495358491487
(%o3) 2 (------------------------- - zeta(3))
1401731326612193601024000
(%i4) psi[2](21);
(%o4) psi (21)
2
Categories: Gamma and factorial functions ·
Option variable: maxpsinegint
Default value: -10
maxpsinegint is the most negative value for which
will try to compute an exact value for rational x. That is if x is less than maxpsinegint, will not return simplified answer, even if it could.
(%i1) psi[0](-100/9);
100
(%o1) psi (- ---)
0 9
(%i2) psi[0](-100/11);
100 %pi 1 5231385863539
(%o2) %pi cot(-------) + psi (--) + -------------
11 0 11 381905105400
(%i3) psi[2](-100/9);
100
(%o3) psi (- ---)
2 9
(%i4) psi[2](-100/11);
3 100 %pi 2 100 %pi 1
(%o4) 2 %pi cot(-------) csc (-------) + psi (--)
11 11 2 11
74191313259470963498957651385614962459
+ --------------------------------------
27850718060013605318710152732000000
Categories: Gamma and factorial functions ·
Option variable: maxpsifracnum
Default value: 6
Let x be a rational number of the form p/q. If p is greater than maxpsifracnum, then
will not try to return a simplified value.
(%i1) psi[0](3/4);
%pi
(%o1) - 3 log(2) + --- - %gamma
2
(%i2) psi[2](3/4);
1 3
(%o2) psi (-) + 4 %pi
2 4
(%i3) maxpsifracnum:2;
(%o3) 2
(%i4) psi[0](3/4);
3
(%o4) psi (-)
0 4
(%i5) psi[2](3/4);
1 3
(%o5) psi (-) + 4 %pi
2 4
Categories: Gamma and factorial functions ·
Option variable: maxpsifracdenom
Default value: 6
Let x be a rational number of the form p/q. If q is greater than maxpsifracdenom, then
will not try to return a simplified value.
(%i1) psi[0](3/4);
%pi
(%o1) - 3 log(2) + --- - %gamma
2
(%i2) psi[2](3/4);
1 3
(%o2) psi (-) + 4 %pi
2 4
(%i3) maxpsifracdenom:2;
(%o3) 2
(%i4) psi[0](3/4);
3
(%o4) psi (-)
0 4
(%i5) psi[2](3/4);
1 3
(%o5) psi (-) + 4 %pi
2 4
Categories: Gamma and factorial functions ·
Function: makefact (expr)
Transforms instances of binomial, gamma, and beta functions in expr into factorials.
See also makegamma.
(%i1) makefact(binomial(n,k));
n!
(%o1) -----------
k! (n - k)!
(%i2) makefact(gamma(x));
(%o2) (x - 1)!
(%i3) makefact(beta(a,b));
(a - 1)! (b - 1)!
(%o3) -----------------
(b + a - 1)!
Categories: Gamma and factorial functions ·
Function: numfactor (expr)
Returns the numerical factor multiplying the expression expr, which should be a single term.
content returns the greatest common divisor (gcd) of all terms in a sum.
(%i1) gamma (7/2);
15 sqrt(%pi)
(%o1) ------------
8
(%i2) numfactor (%);
15
(%o2) --
8
Categories: Expressions ·
Next: Error Function, Previous: Gamma and Factorial Functions, Up: Special Functions [Contents][Index]
17.5 Exponential Integrals
The Exponential Integral and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 5.
Function: expintegral_e1 (z)
The Exponential Integral E1(z) defined as
with
(A&S eqn 5.1.1) and (DLMF 6.2E2)
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_ei (x)
The Exponential Integral Ei(x) defined as
with x real and x > 0. (A&S eqn 5.1.2) and (DLMF 6.2E5)
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_li (x)
The Exponential Integral li(x) defined as
with x real and x > 1. (A&S eqn 5.1.3) and (DLMF 6.2E8)
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_e (n,z)
The Exponential Integral En(z) (A&S eqn 5.1.4) defined as
with
and n a non-negative integer.
For half-integral orders, this can be written in terms of erfc or erf. See expintexpand for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_si (z)
The Exponential Integral Si(z) (A&S eqn 5.2.1 and DLMF 6.2#E9) defined as
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_ci (z)
The Exponential Integral Ci(z) (A&S eqn 5.2.2 and DLMF 6.2#E13) defined as
with
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_shi (z)
The Exponential Integral Shi(z) (A&S eqn 5.2.3 and DLMF 6.2#E15) defined as
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Function: expintegral_chi (z)
The Exponential Integral Chi(z) (A&S eqn 5.2.4 and DLMF 6.2#E16) defined as
with
This can be written in terms of other functions. See expintrep for examples.
Categories: Exponential Integrals · Special functions ·
Option variable: expintrep
Default value: false
Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.
Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.
false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci; and expintegral_hyp means expintegral_shi or expintegral_chi.
Here are some examples for expintrep set to gamma_incomplete:
(%i1) expintrep:'gamma_incomplete;
(%o1) gamma_incomplete
(%i2) expintegral_e1(z);
(%o2) gamma_incomplete(0, z)
(%i3) expintegral_ei(z);
(%o3) log(z) - log(- z) - gamma_incomplete(0, - z)
(%i4) expintegral_li(z);
(%o4) log(log(z)) - log(- log(z)) - gamma_incomplete(0, - log(z))
(%i5) expintegral_e(n,z);
n - 1
(%o5) gamma_incomplete(1 - n, z) z
(%i6) expintegral_si(z);
(%o6) (%i (- log(%i z) + log(- %i z) - gamma_incomplete(0, %i z)
+ gamma_incomplete(0, - %i z)))/2
(%i7) expintegral_ci(z);
(%o7) log(z) - (log(%i z) + log(- %i z)
+ gamma_incomplete(0, %i z) + gamma_incomplete(0, - %i z))/2
(%i8) expintegral_shi(z);
(%o8) (log(z) - log(- z) + gamma_incomplete(0, z)
- gamma_incomplete(0, - z))/2
(%i9) expintegral_chi(z);
(%o9) - (- log(z) + log(- z) + gamma_incomplete(0, z)
+ gamma_incomplete(0, - z))/2
For expintrep set to expintegral_e1:
(%i1) expintrep:'expintegral_e1;
(%o1) expintegral_e1
(%i2) expintegral_ei(z);
(%o2) log(z) - log(- z) - expintegral_e1(- z)
(%i3) expintegral_li(z);
(%o3) log(log(z)) - log(- log(z)) - expintegral_e1(- log(z))
(%i4) expintegral_e(n,z);
(%o4) expintegral_e(n, z)
(%i5) expintegral_si(z);
(%o5) (%i (- log(%i z) - expintegral_e1(%i z) + log(- %i z)
+ expintegral_e1(- %i z)))/2
(%i6) expintegral_ci(z);
(%o6) log(z) - (log(- %i z) (expintegral_e1(%i z)
+ expintegral_e1(- %i z)) log(%i z))/2
(%i7) expintegral_shi(z);
log(z) + expintegral_e1(z) - log(- z) - expintegral_e1(- z)
(%o7) -----------------------------------------------------------
2
(%i8) expintegral_chi(z);
(%o8)
- log(z) + expintegral_e1(z) + log(- z) + expintegral_e1(- z)
- -------------------------------------------------------------
2
For expintrep set to expintegral_ei:
(%i1) expintrep:'expintegral_ei;
(%o1) expintegral_ei
(%i2) expintegral_e1(z);
1
log(- z) - log(- -)
z
(%o2) - log(z) + ------------------- - expintegral_ei(- z)
2
(%i3) expintegral_ei(z);
(%o3) expintegral_ei(z)
(%i4) expintegral_li(z);
(%o4) expintegral_ei(log(z))
(%i5) expintegral_e(n,z);
(%o5) expintegral_e(n, z)
(%i6) expintegral_si(z);
(%o6) (%i (log(%i z) + 2 (expintegral_ei(- %i z)
%i %i
- expintegral_ei(%i z)) - log(- %i z) + log(--) - log(- --)))/4
z z
(%i7) expintegral_ci(z);
(%o7) (- log(%i z) + 2 (expintegral_ei(%i z)
%i %i
+ expintegral_ei(- %i z)) - log(- %i z) + log(--) + log(- --))/4
z z
+ log(z)
(%i8) expintegral_shi(z);
(%o8) (- 2 log(z) + 2 (expintegral_ei(z) - expintegral_ei(- z))
1
+ log(- z) - log(- -))/4
z
(%i9) expintegral_chi(z);
(%o9) (2 log(z) + 2 (expintegral_ei(z) + expintegral_ei(- z))
1
- log(- z) + log(- -))/4
z
For expintrep set to expintegral_li:
(%i1) expintrep:'expintegral_li;
(%o1) expintegral_li
(%i2) expintegral_e1(z);
1
log(- z) - log(- -)
- z z
(%o2) - expintegral_li(%e ) - log(z) + -------------------
2
(%i3) expintegral_ei(z);
z
(%o3) expintegral_li(%e )
(%i4) expintegral_li(z);
(%o4) expintegral_li(z)
(%i5) expintegral_e(n,z);
(%o5) expintegral_e(n, z)
(%i6) expintegral_si(z);
%i z - %e z
(%o6) - (%i (expintegral_li(%e ) - expintegral_li(%e )
%pi signum(z)
- -------------))/2
2
(%i7) expintegral_ci(z);
%i z - %i z
expintegral_li(%e ) + expintegral_li(%e )
(%o7) -------------------------------------------------
2
- signum(z) + 1
(%i8) expintegral_shi(z);
z - z
expintegral_li(%e ) - expintegral_li(%e )
(%o8) -------------------------------------------
2
(%i9) expintegral_chi(z);
z - z
expintegral_li(%e ) + expintegral_li(%e )
(%o9) -------------------------------------------
2
For expintrep set to expintegral_trig:
(%i1) expintrep:'expintegral_trig;
(%o1) expintegral_trig
(%i2) expintegral_e1(z);
(%o2) log(%i z) - %i expintegral_si(%i z) - expintegral_ci(%i z)
- log(z)
(%i3) expintegral_ei(z);
(%o3) - log(%i z) - %i expintegral_si(%i z)
+ expintegral_ci(%i z) + log(z)
(%i4) expintegral_li(z);
(%o4) - log(%i log(z)) - %i expintegral_si(%i log(z))
+ expintegral_ci(%i log(z)) + log(log(z))
(%i5) expintegral_e(n,z);
(%o5) expintegral_e(n, z)
(%i6) expintegral_si(z);
(%o6) expintegral_si(z)
(%i7) expintegral_ci(z);
(%o7) expintegral_ci(z)
(%i8) expintegral_shi(z);
(%o8) - %i expintegral_si(%i z)
(%i9) expintegral_chi(z);
(%o9) - log(%i z) + expintegral_ci(%i z) + log(z)
For expintrep set to expintegral_hyp:
(%i1) expintrep:'expintegral_hyp;
(%o1) expintegral_hyp
(%i2) expintegral_e1(z);
(%o2) expintegral_shi(z) - expintegral_chi(z)
(%i3) expintegral_ei(z);
(%o3) expintegral_shi(z) + expintegral_chi(z)
(%i4) expintegral_li(z);
(%o4) expintegral_shi(log(z)) + expintegral_chi(log(z))
(%i5) expintegral_e(n,z);
(%o5) expintegral_e(n, z)
(%i6) expintegral_si(z);
(%o6) - %i expintegral_shi(%i z)
(%i7) expintegral_ci(z);
(%o7) - log(%i z) + expintegral_chi(%i z) + log(z)
(%i8) expintegral_shi(z);
(%o8) expintegral_shi(z)
(%i9) expintegral_chi(z);
(%o9) expintegral_chi(z)
Categories: Exponential Integrals ·
Option variable: expintexpand
Default value: false
Expand expintegral_e(n,z) for half integral values in terms of erfc or erf and for positive integers in terms of expintegral_ei.
(%i1) expintegral_e(1/2,z);
1
(%o1) expintegral_e(-, z)
2
(%i2) expintegral_e(1,z);
(%o2) expintegral_e(1, z)
(%i3) expintexpand:true;
(%o3) true
(%i4) expintegral_e(1/2,z);
sqrt(%pi) erfc(sqrt(z))
(%o4) -----------------------
sqrt(z)
(%i5) expintegral_e(1,z);
1
log(- -) - log(- z)
z
(%o5) - log(z) - ------------------- - expintegral_ei(- z)
2
Categories: Exponential Integrals ·
Next: Struve Functions, Previous: Exponential Integrals, Up: Special Functions [Contents][Index]