Next: Elliptic Functions, Previous: Polynomials [Contents][Index]
Next: Bessel Functions, Up: Special Functions [Contents][Index]
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
Next: Airy Functions, Previous: Introduction to Special Functions, Up: Special Functions [Contents][Index]
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.
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
$$
Y_v(z) = {{\cos(\pi v)\, J_v(z) - J_{-v}(z)}\over{\sin{\pi v}}}
$$
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.
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
$$
I_v(z) = \sum_{k=0}^{\infty } {{1\over{k!\,\Gamma
\left(v+k+1\right)}} {\left(z\over 2\right)^{v+2\,k}}}
$$
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.
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
$$
K_v(z) = {1\over 2} \pi\, {I_{-v}(z)-I_{v}(z) \over \sin v\pi}
$$
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.
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) 0.24226845767487 - 1.471472392670243 %i
(%i2) hankel_1(1,0.5+%i); (%o2) - 0.25582879948621 %i - 0.23957560188301
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
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.
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)
The scaled modified Bessel function of the first kind of order v and argument z. That is,
$$ {\rm scaled\_bessel\_i}(v,z) = e^{-|z|} I_v(z). $$This function is particularly useful
for calculating
\(I_v(z)\)
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).
Identical to scaled_bessel_i(0,z).
Identical to scaled_bessel_i(1,z).
Lommel’s little \(s_{\mu,\nu}(z)\) function. (DLMF 11.9.E3)(G&R 8.570.1).
This Lommel function is the particular solution of the inhomogeneous Bessel differential equation:
$$ {d^2\over dz^2} + {1\over z}{dw\over dz} + \left(1-{\nu^2\over z^2}\right) w = z^{\mu-1} $$This can be defined by the series
$$ s_{\mu,\nu}(z) = z^{\mu+1}\sum_{k=0}^{\infty} (-1)^k {z^{2k}\over a_{k+1}(\mu, \nu)} $$where
$$ a_k(\mu,\nu) = \prod_{m=1}^k \left(\left(\mu + 2m-1\right)^2-\nu^2\right) = 4^k\left(\mu-\nu+1\over 2\right)_k \left(\mu+\nu+1\over 2\right)_k $$Lommel’s big \(S_{\mu,\nu}(z)\) 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 \(\mu\) and \(\nu,\) where
$$ \eqalign{ S_{\mu,\nu}(z) = s_{\mu,\nu}(z) + 2^{\mu-1} & \Gamma\left({\mu\over 2} + {\nu\over 2} + {1\over 2}\right) \Gamma\left({\mu\over 2} - {\nu\over 2} + {1\over 2}\right) \cr & \times \left(\sin\left({(\mu-\nu)\pi\over 2}\right) J_{\nu}(z) - \cos\left({(\mu-\nu)\pi\over 2}\right) Y_{\nu}(z)\right) } $$When \(\mu\pm \nu\) is an odd negative integer, the limit must be used.
Next: Gamma and Factorial Functions, Previous: Bessel Functions, Up: Special Functions [Contents][Index]
The Airy functions \({\rm Ai}(x)\) and \({\rm Bi}(x)\) are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Section 10.4 and DLMF 9.
The Airy differential equation is:
$$ {d^2 y\over dx^2} - xy = 0 $$The numerically satisfactory pair of solutions (DLMF 9.2#T1) on the real line are \(y = {\rm Ai}(x)\) and \(y = {\rm Bi}(x).\) These two solutions are oscillatory for x < 0. \({\rm Ai}(x)\) is the solution subject to the condition that \(y\rightarrow 0\) as \(x\rightarrow +\infty,\) and \({\rm Bi}(x)\) is the second solution with the same amplitude as \({\rm Ai}(x)\) as \(x\rightarrow-\infty\) which differs in phase by \(\pi/2.\) Also, \({\rm Bi}(x)\) is unbounded as \(x\rightarrow +\infty.\)
If the argument x is a real or complex floating point number, the numerical value of the function is returned.
The Airy function \({\rm Ai}(x).\) See A&S eqn 10.4.2 and DLMF 9.
See also airy_bi, airy_dai, and airy_dbi.
The derivative of the Airy function \({\rm Ai}(x):\)
$$ {\rm airy\_dai}(x) = {d\over dx}{\rm Ai}(x) $$See airy_ai.
The Airy function \({\rm Bi}(x).\) See A&S eqn 10.4.3 and DLMF 9.
The derivative of the Airy function \({\rm Bi}(x):\)
$$ {\rm airy\_dbi}(x) = {d\over dx}{\rm Bi}(x) $$Next: Exponential Integrals, Previous: Airy Functions, Up: Special Functions [Contents][Index]
The gamma function and the related beta, psi and incomplete gamma functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Chapter 6.
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
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
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
The basic definition of the gamma function (DLMF 5.2.E1 and A&S eqn 6.1.1) is
$$ \Gamma\left(z\right)=\int_{0}^{\infty }{t^{z-1}\,e^ {- t }\;dt} $$Maxima simplifies gamma for positive integer and positive and negative
rational numbers. For half integral values the result is a rational number
times
\(\sqrt{\pi}.\)
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:
(%i4) map('gamma,[2.5,2.5b0]);
(%o4) [1.329340388179137, 1.3293403881791370205b0]
(%i5) map('gamma,[1.0+%i,1.0b0+%i]);
(%o5) [0.498015668118356 - .1549498283018107 %i,
4.9801566811835604272b-1 - 1.5494982830181068513b-1 %i]
gamma has mirror symmetry:
(%i6) declare(z,complex)$ (%i7) conjugate(gamma(z)); (%o7) gamma(conjugate(z))
Maxima expands gamma(z+n) and gamma(z-n), when gamma_expand
is true:
(%i8) gamma_expand:true$
(%i9) [gamma(z+1),gamma(z-1),gamma(z+2)/gamma(z+1)];
gamma(z)
(%o9) [z gamma(z), --------, z + 1]
z - 1
The derivative of gamma:
(%i10) diff(gamma(z),z);
(%o10) psi (z) gamma(z)
0
See also makegamma.
The Euler-Mascheroni constant is %gamma.
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
The lower incomplete gamma function (DLMF 8.2.E1 and A&S eqn 6.5.2):
$$ \gamma\left(a , z\right)=\int_{0}^{z}{t^{a-1}\,e^ {- t }\;dt} $$See also gamma_incomplete (upper incomplete gamma function).
The incomplete upper gamma function (DLMF 8.2.E2 and A&S eqn 6.5.3):
$$ \Gamma\left(a , z\right)=\int_{z}^{\infty }{t^{a-1}\,e^ {- t }\;dt} $$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.
The regularized incomplete upper gamma function (DLMF 8.2.E4):
$$ Q\left(a , z\right)={{\Gamma\left(a , z\right)}\over{\Gamma\left(a\right)}} $$See also gamma_expand for controlling how
gamma_incomplete is expressed in terms of elementary functions
and erfc.
Also see gamma_incomplete.
The generalized incomplete gamma function.
$$ \Gamma\left(a , z_{1}, z_{2}\right)=\int_{z_{1}}^{z_{2}}{t^{a-1}\,e^ {- t }\;dt} $$Also see gamma_incomplete and gamma_incomplete_regularized.
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 - z
(%o7) ---------------------- - z (--------------- + -----) %e
(1 - a) (2 - a) (a - 2) (a - 1) a - 2
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.
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)
The beta function is defined as $$ {\rm B}(a, b) = {{\Gamma(a) \Gamma(b)}\over{\Gamma(a+b)}} $$
(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:
(%i2) [beta(1/2,5/2),beta(1/3,2/3),beta(1/4,3/4)];
3 %pi 2 %pi
(%o2) [-----, -------, 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:
(%i3) beta_args_sum_to_integer:true$
(%i4) beta(a+1,-a+2);
%pi (a - 1) a
(%o4) ------------------
2 sin(%pi (2 - a))
The possible results, when one of the arguments is a negative integer:
(%i5) [beta(-3,1),beta(-3,2),beta(-3,3)];
1 1 1
(%o5) [- -, -, - -]
3 6 3
beta(a+n,b) or beta(a-n,b) with n an integer simplifies when
beta_expand is true:
(%i6) beta_expand:true$
(%i7) [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
(%o7) [------------, ----------------------, -]
b + a a - 1 b
Beta is not defined, when one of the arguments is zero:
(%i7) beta(0,b); beta: expected nonzero arguments; found 0, b -- an error. To debug this try debugmode(true);
Numerical evaluation for real and complex arguments in float or bigfloat precision:
(%i8) beta(2.5,2.3); (%o8) .08694748611299981 (%i9) beta(2.5,1.4+%i); (%o9) 0.0640144950796695 - .1502078053286415 %i (%i10) beta(2.5b0,2.3b0); (%o10) 8.694748611299969b-2 (%i11) beta(2.5b0,1.4b0+%i); (%o11) 6.401449507966944b-2 - 1.502078053286415b-1 %i
Beta is symmetric and has mirror symmetry:
(%i14) beta(a,b)-beta(b,a); (%o14) 0 (%i15) declare(a,complex,b,complex)$ (%i16) conjugate(beta(a,b)); (%o16) beta(conjugate(a), conjugate(b))
The derivative of the beta function wrt a:
(%i17) diff(beta(a,b),a);
(%o17) - beta(a, b) (psi (b + a) - psi (a))
0 0
The basic definition of the incomplete beta function (DLMF 8.17.E1 and A&S eqn 6.6.1) is
$$ {\rm B}_z(a,b) = \int_0^z t^{a-1}(1-t)^{b-1}\; dt $$This definition is possible for \({\rm Re}(a) > 0\) and \({\rm Re}(b) > 0\) and \(|z| < 1.\) 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 \(m \le n\) the incomplete beta function is defined through
$$ z^{n-1}\sum_{k=0}^{m-1} {{(1-m)_k z^k} \over {k! (n-k)}} $$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
\({\rm Re}(a) > 0,\)
beta_incomplete has the
specific value zero. For z=1 and
\({\rm Re}(b) > 0,\)
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:
(%i2) beta_incomplete(a,2,z);
a
(a (1 - z) + 1) z
(%o2) ------------------
a (a + 1)
Simplification for a and b a positive integer:
(%i3) beta_incomplete(3,2,z);
3
(3 (1 - z) + 1) z
(%o3) ------------------
12
a is a negative integer and b<=(-a), Maxima simplifies:
(%i4) beta_incomplete(-3,1,z);
1
(%o4) - ----
3
3 z
For the specific values z=0 and z=1, Maxima simplifies:
(%i5) assume(a>0,b>0)$ (%i6) beta_incomplete(a,b,0); (%o6) 0 (%i7) beta_incomplete(a,b,1); (%o7) beta(a, b)
Numerical evaluation in float or bigfloat precision:
(%i8) beta_incomplete(0.25,0.50,0.9); (%o8) 4.594959440269333 (%i9) fpprec:25$ (%i10) beta_incomplete(0.25,0.50,0.9b0); (%o10) 4.594959440269324086971203b0
For abs(z)>1 beta_incomplete returns a complex result:
(%i11) beta_incomplete(0.25,0.50,1.7); (%o11) 5.244115108584249 - 1.45518047787844 %i
Results for more general complex arguments:
(%i14) beta_incomplete(0.25+%i,1.0+%i,1.7+%i); (%o14) 2.726960675662536 - .3831175704269199 %i (%i15) beta_incomplete(1/2,5/4*%i,2.8+%i); (%o15) 13.04649635168716 %i - 5.802067956270001 (%i16)
Expansion, when beta_expand is true:
(%i23) beta_incomplete(a+1,b,z),beta_expand:true;
b a
a beta_incomplete(a, b, z) (1 - z) z
(%o23) -------------------------- - -----------
b + a b + a
(%i24) beta_incomplete(a-1,b,z),beta_expand:true;
b a - 1
beta_incomplete(a, b, z) (- b - a + 1) (1 - z) z
(%o24) -------------------------------------- - ---------------
1 - a 1 - a
Derivative and integral for beta_incomplete:
(%i34) diff(beta_incomplete(a, b, z), z);
b - 1 a - 1 (%o34) (1 - z) z
(%i35) integrate(beta_incomplete(a, b, z), z);
b a
(1 - z) z
(%o35) ----------- + beta_incomplete(a, b, z) z
b + a
a beta_incomplete(a, b, z)
- --------------------------
b + a
(%i36) factor(diff(%, z));
(%o36) beta_incomplete(a, b, z)
The regularized incomplete beta function (DLMF 8.17.E2 and A&S eqn 6.6.2), defined as
$$ I_z(a,b) = {{\rm B}_z(a,b)\over {\rm B}(a,b)} $$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
\({\rm Re}(a)>0,\)
beta_incomplete_regularized has
the specific value 0. For z=1 and
\({\rm Re}(b) > 0,\)
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:
(%i4) assume(a>0,b>0)$ (%i5) beta_incomplete_regularized(a,b,0); (%o5) 0 (%i6) beta_incomplete_regularized(a,b,1); (%o6) 1
Numerical evaluation for real and complex arguments in float and bigfloat precision:
(%i7) beta_incomplete_regularized(0.12,0.43,0.9); (%o7) .9114011367359802 (%i8) fpprec:32$ (%i9) beta_incomplete_regularized(0.12,0.43,0.9b0); (%o9) 9.1140113673598075519946998779975b-1 (%i10) beta_incomplete_regularized(1+%i,3/3,1.5*%i); (%o10) .2865367499935403 %i - 0.122995963334684 (%i11) fpprec:20$ (%i12) beta_incomplete_regularized(1+%i,3/3,1.5b0*%i); (%o12) 2.8653674999354036142b-1 %i - 1.2299596333468400163b-1
Expansion, when beta_expand is true:
(%i13) beta_incomplete_regularized(a+1,b,z);
b a
(1 - z) z
(%o13) beta_incomplete_regularized(a, b, z) - ------------
a beta(a, b)
(%i14) beta_incomplete_regularized(a-1,b,z);
(%o14) beta_incomplete_regularized(a, b, z)
b a - 1
(1 - z) z
- ----------------------
beta(a, b) (b + a - 1)
The derivative and the integral wrt z:
(%i15) diff(beta_incomplete_regularized(a,b,z),z);
b - 1 a - 1
(1 - z) z
(%o15) -------------------
beta(a, b)
(%i16) integrate(beta_incomplete_regularized(a,b,z),z);
(%o16) beta_incomplete_regularized(a, b, z) z
b a
(1 - z) z
a (beta_incomplete_regularized(a, b, z) - ------------)
a beta(a, b)
- -------------------------------------------------------
b + a
The basic definition of the generalized incomplete beta function is
$$ \int_{z_1}^{z_2} t^{a-1}(1-t)^{b-1}\; dt $$Maxima simplifies beta_incomplete_regularized for a and b
a positive integer.
For
\({\rm Re}(a) > 0\)
and
\(z_1 = 0\)
or
\(z_2 = 0,\)
Maxima simplifies
beta_incomplete_generalized to beta_incomplete.
For
\({\rm Re}(b) > 0\)
and
\(z_1 = 1\)
or
\(z_2 = 1,\)
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:
(%i4) assume(a > 0, b > 0)$ (%i5) beta_incomplete_generalized(a,b,z1,0); (%o5) - beta_incomplete(a, b, z1) (%i6) beta_incomplete_generalized(a,b,0,z2); (%o6) - beta_incomplete(a, b, z2) (%i7) beta_incomplete_generalized(a,b,z1,1); (%o7) beta(a, b) - beta_incomplete(a, b, z1) (%i8) beta_incomplete_generalized(a,b,1,z2); (%o8) beta_incomplete(a, b, z2) - beta(a, b)
Numerical evaluation for real arguments in float or bigfloat precision:
(%i9) beta_incomplete_generalized(1/2,3/2,0.25,0.31); (%o9) .09638178086368676 (%i10) fpprec:32$ (%i10) beta_incomplete_generalized(1/2,3/2,0.25,0.31b0); (%o10) 9.6381780863686935309170054689964b-2
Numerical evaluation for complex arguments in float or bigfloat precision:
(%i11) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31); (%o11) - .09625463003205376 %i - .003323847735353769 (%i12) fpprec:20$ (%i13) beta_incomplete_generalized(1/2+%i,3/2+%i,0.25,0.31b0); (%o13) - 9.6254630032054178691b-2 %i - 3.3238477353543591914b-3
Expansion for a+n or a-n, n a positive integer, when
beta_expand is true:
(%i14) beta_expand:true$
(%i15) beta_incomplete_generalized(a+1,b,z1,z2);
b a b a
(1 - z1) z1 - (1 - z2) z2
(%o15) -----------------------------
b + a
a beta_incomplete_generalized(a, b, z1, z2)
+ -------------------------------------------
b + a
(%i16) beta_incomplete_generalized(a-1,b,z1,z2);
beta_incomplete_generalized(a, b, z1, z2) (- b - a + 1)
(%o16) -------------------------------------------------------
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:
(%i17) diff(beta_incomplete_generalized(a,b,z1,z2),z1);
b - 1 a - 1
(%o17) - (1 - z1) z1
(%i18) integrate(beta_incomplete_generalized(a,b,z1,z2),z1);
(%o18) beta_incomplete_generalized(a, b, z1, z2) z1
+ beta_incomplete(a + 1, b, z1)
(%i19) integrate(beta_incomplete_generalized(a,b,z1,z2),z2);
(%o19) beta_incomplete_generalized(a, b, z1, z2) z2
- beta_incomplete(a + 1, b, z2)
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.
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.
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
$$
\psi^{(n)}(x) = {d^{n+1}\over{dx^{n+1}}} \log\Gamma(x)
$$
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
\(\psi^{(n)}(x)\)
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,
\(\psi^{(0)}(x)\)
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) - 5.30263321633764
(%i5) psi[2](3/4);
1 3
(%o5) psi (-) + 4 %pi
2 4
(%i6) float(%);
(%o6) - 5.30263321633764
Default value: 20
maxpsiposint is the largest positive integer value for
which
\(\psi^{(n)}(m)\)
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
Default value: -10
maxpsinegint is the most negative value for
which
\(\psi^{(0)}(x)\)
will try to compute an exact
value for rational x. That is if x is less than
maxpsinegint,
\(\psi^{(n)}(x)\)
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
Default value: 6
Let x be a rational number of the form p/q.
If p is greater than maxpsifracnum,
then
\(\psi^{(0)}(x)\)
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
Default value: 6
Let x be a rational number of the form p/q.
If q is greater than maxpsifracdenom,
then
\(\psi^{(0)}(x)\)
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
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)!
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
Next: Error Function, Previous: Gamma and Factorial Functions, Up: Special Functions [Contents][Index]
The Exponential Integral and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 5.
The Exponential Integral E1(z) defined as
$$ E_1(z) = \int_z^\infty {e^{-t} \over t} dt $$with \(\left| \arg z \right| < \pi.\) (A&S eqn 5.1.1) and (DLMF 6.2E2)
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Ei(x) defined as
$$ Ei(x) = - -\kern-10.5pt\int_{-x}^\infty {e^{-t} \over t} dt = -\kern-10.5pt\int_{-\infty}^x {e^{t} \over t} dt $$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.
The Exponential Integral li(x) defined as
$$ li(x) = -\kern-10.5pt\int_0^x {dt \over \ln t} $$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.
The Exponential Integral En(z) (A&S eqn 5.1.4) defined as
$$ E_n(z) = \int_1^\infty {e^{-zt} \over t^n} dt $$with \({\rm Re}(z) > 1\) and n a non-negative integer.
For half-integral orders, this can be written in terms of erfc
or erf. See expintexpand for examples.
The Exponential Integral Si(z) (A&S eqn 5.2.1 and DLMF 6.2#E9) defined as
$$ {\rm Si}(z) = \int_0^z {\sin t \over t} dt $$This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Ci(z) (A&S eqn 5.2.2 and DLMF 6.2#E13) defined as
$$ {\rm Ci}(z) = \gamma + \log z + \int_0^z {{\cos t - 1} \over t} dt $$with \(|\arg z| < \pi.\)
This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Shi(z) (A&S eqn 5.2.3 and DLMF 6.2#E15) defined as
$$ {\rm Shi}(z) = \int_0^z {\sinh t \over t} dt $$This can be written in terms of other functions. See expintrep for examples.
The Exponential Integral Chi(z) (A&S eqn 5.2.4 and DLMF 6.2#E16) defined as
$$ {\rm Chi}(z) = \gamma + \log z + \int_0^z {{\cosh t - 1} \over t} dt $$with \(|\arg z| < \pi.\)
This can be written in terms of other functions. See expintrep for examples.
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);
log(z) - log(- z) + gamma_incomplete(0, z) - gamma_incomplete(0, - z)
(%o8) ---------------------------------------------------------------------
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);
(- log(z)) + expintegral_e1(z) + log(- z) + expintegral_e1(- z)
(%o8) - ---------------------------------------------------------------
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) - expintegral_ei(%i z))
%i %i
- log(- %i z) + log(--) - log(- --)))/4
z z
(%i7) expintegral_ci(z);
(%o7) ((- log(%i z)) + 2 (expintegral_ei(%i z) + expintegral_ei(- %i z))
%i %i
- log(- %i z) + log(--) + log(- --))/4 + log(z)
z z
(%i8) expintegral_shi(z);
(%o8) ((- 2 log(z)) + 2 (expintegral_ei(z) - expintegral_ei(- z)) + log(- z)
1
- log(- -))/4
z
(%i9) expintegral_chi(z);
(%o9)
1
2 log(z) + 2 (expintegral_ei(z) + expintegral_ei(- z)) - log(- z) + log(- -)
z
----------------------------------------------------------------------------
4
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 %pi signum(z)
%i (expintegral_li(%e ) - expintegral_li(%e ) - -------------)
2
(%o6) - ----------------------------------------------------------------------
2
(%i7) expintegral_ci(z);
%i z - %i z
expintegral_li(%e ) + expintegral_li(%e )
(%o7) ------------------------------------------------- - signum(z) + 1
2
(%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)
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
Next: Struve Functions, Previous: Exponential Integrals, Up: Special Functions [Contents][Index]
The Error function and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 7 and (DLMF 7)
The Error Function erf(z): $$ {\rm erf}\ z = {{2\over \sqrt{\pi}}} \int_0^z e^{-t^2}\, dt $$
(A&S eqn 7.1.1) and (DLMF 7.2.E1).
See also flag erfflag. This can also be expressed in terms
of a hypergeometric function. See hypergeometric_representation.
The Complementary Error Function erfc(z): $$ {\rm erfc}\ z = 1 - {\rm erf}\ z $$
(A&S eqn 7.1.2) and (DLMF 7.2.E2).
This can also be expressed in terms of a hypergeometric function. See hypergeometric_representation.
The Imaginary Error Function. $$ {\rm erfi}\ z = -i\, {\rm erf}(i z) $$
Generalized Error function Erf(z1,z2): $$ {\rm erf}(z_1, z_2) = {{2\over \sqrt{\pi}}} \int_{z_1}^{z_2} e^{-t^2}\, dt $$
This can also be expressed in terms of a hypergeometric function. See hypergeometric_representation.
The Fresnel Integral
$$ C(z) = \int_0^z \cos\left({\pi \over 2} t^2\right)\, dt $$(A&S eqn 7.3.1) and (DLMF 7.2.E7).
The simplification
\(C(-x) = -C(x)\)
is applied when
flag trigsign is true.
The simplification
\(C(ix) = iC(x)\)
is applied when
flag %iargs is true.
See flags erf_representation and hypergeometric_representation.
The Fresnel Integral $$ S(z) = \int_0^z \sin\left({\pi \over 2} t^2\right)\, dt $$
(A&S eqn 7.3.2) and (DLMF 7.2.E8).
The simplification
\(S(-x) = -S(x)\)
is applied when
flag trigsign is true.
The simplification
\(S(ix) = iS(x)\)
is applied when
flag %iargs is true.
See flags erf_representation and hypergeometric_representation.
Default value: false
erf_representation controls how the error functions are
represented. It must be set to one of false, erf,
erfc, or erfi. When set to false, the error functions are not
modified. When set to erf, all error functions (erfc,
erfi, erf_generalized, fresnel_s and
fresnel_c) are converted to erf functions. Similary,
erfc converts error functions to erfc. Finally
erfi converts the functions to erfi.
Converting to erf:
(%i1) erf_representation:erf;
(%o1) true
(%i2) erfc(z);
(%o2) erfc(z)
(%i3) erfi(z);
(%o3) erfi(z)
(%i4) erf_generalized(z1,z2);
(%o4) erf(z2) - erf(z1)
(%i5) fresnel_c(z);
sqrt(%pi) (%i + 1) z sqrt(%pi) (1 - %i) z
(1 - %i) (erf(--------------------) + %i erf(--------------------))
2 2
(%o5) -------------------------------------------------------------------
4
(%i6) fresnel_s(z);
sqrt(%pi) (%i + 1) z sqrt(%pi) (1 - %i) z
(%i + 1) (erf(--------------------) - %i erf(--------------------))
2 2
(%o6) -------------------------------------------------------------------
4
Converting to erfc:
(%i1) erf_representation:erfc;
(%o1) erfc
(%i2) erf(z);
(%o2) 1 - erfc(z)
(%i3) erfc(z);
(%o3) erfc(z)
(%i4) erf_generalized(z1,z2);
(%o4) erfc(z1) - erfc(z2)
(%i5) fresnel_s(c);
sqrt(%pi) (%i + 1) c
(%o5) ((%i + 1) ((- erfc(--------------------))
2
sqrt(%pi) (1 - %i) c
- %i (1 - erfc(--------------------)) + 1))/4
2
(%i6) fresnel_c(c);
sqrt(%pi) (%i + 1) c
(%o6) ((1 - %i) ((- erfc(--------------------))
2
sqrt(%pi) (1 - %i) c
+ %i (1 - erfc(--------------------)) + 1))/4
2
Converting to erfc:
(%i1) erf_representation:erfi;
(%o1) erfi
(%i2) erf(z);
(%o2) - %i erfi(%i z)
(%i3) erfc(z);
(%o3) %i erfi(%i z) + 1
(%i4) erfi(z);
(%o4) erfi(z)
(%i5) erf_generalized(z1,z2);
(%o5) %i erfi(%i z1) - %i erfi(%i z2)
(%i6) fresnel_s(z);
sqrt(%pi) %i (%i + 1) z
(%o6) ((%i + 1) ((- %i erfi(-----------------------))
2
sqrt(%pi) (1 - %i) %i z
- erfi(-----------------------)))/4
2
(%i7) fresnel_c(z);
(%o7)
sqrt(%pi) (1 - %i) %i z sqrt(%pi) %i (%i + 1) z
(1 - %i) (erfi(-----------------------) - %i erfi(-----------------------))
2 2
---------------------------------------------------------------------------
4
Default value: false
Enables transformation to a Hypergeometric
representation for fresnel_s and fresnel_c and other
error functions.
(%i1) hypergeometric_representation:true;
(%o1) true
(%i2) fresnel_s(z);
2 4
3 3 7 %pi z 3
%pi hypergeometric([-], [-, -], - -------) z
4 2 4 16
(%o2) ---------------------------------------------
6
(%i3) fresnel_c(z);
2 4
1 1 5 %pi z
(%o3) hypergeometric([-], [-, -], - -------) z
4 2 4 16
(%i4) erf(z);
1 3 2
2 hypergeometric([-], [-], - z ) z
2 2
(%o4) ----------------------------------
sqrt(%pi)
(%i5) erfi(z);
1 3 2
2 hypergeometric([-], [-], z ) z
2 2
(%o5) --------------------------------
sqrt(%pi)
(%i6) erfc(z);
1 3 2
2 hypergeometric([-], [-], - z ) z
2 2
(%o6) 1 - ----------------------------------
sqrt(%pi)
(%i7) erf_generalized(z1,z2);
1 3 2
2 hypergeometric([-], [-], - z2 ) z2
2 2
(%o7) ------------------------------------
sqrt(%pi)
1 3 2
2 hypergeometric([-], [-], - z1 ) z1
2 2
- ------------------------------------
sqrt(%pi)
Next: Hypergeometric Functions, Previous: Error Function, Up: Special Functions [Contents][Index]
The Struve functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 12 and (DLMF 11). The Struve Function \({\bf H}_{\nu}(z)\) is a particular solution of the differential equation: $$ z^2 {d^2 w \over dz^2} + z {dw \over dz} + (z^2-\nu^2)w = {{4\left({1\over 2} z\right)^{\nu+1}} \over \sqrt{\pi} \Gamma\left(\nu + {1\over 2}\right)} $$
which has the general soution $$ w = aJ_{\nu}(z) + bY_{\nu}(z) + {\bf H}_{\nu}(z) $$
The Struve Function H of order \(\nu\) and argument z:
$$ {\bf H}_{\nu}(z) = \left({z\over 2}\right)^{\nu+1} \sum_{k=0}^{\infty} {(-1)^k\left({z\over 2}\right)^{2k} \over \Gamma\left(k + {3\over 2}\right) \Gamma\left(k + \nu + {3\over 2}\right)} $$(A&S eqn 12.1.3) and (DLMF 11.2.E1).
When besselexpand is true, struve_h is expanded in terms
of elementary functions when the order v is half of an odd integer.
See besselexpand.
The Modified Struve Function L of order \(\nu\) and argument z: $$ {\bf L}_{\nu}(z) = -ie^{-{i\nu\pi\over 2}} {\bf H}_{\nu}(iz) $$
(A&S eqn 12.2.1) and (DLMF 11.2.E2).
When besselexpand is true, struve_l is expanded in terms
of elementary functions when the order v is half of an odd integer.
See besselexpand.
Next: Parabolic Cylinder Functions, Previous: Struve Functions, Up: Special Functions [Contents][Index]
The Hypergeometric Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapters 13 and A&S 15.
Maxima has very limited knowledge of these functions. They
can be returned from function hgfred.
Whittaker M function (A&S eqn 13.1.32):
$$ M_{\kappa,\mu}(z) = e^{-{1\over 2}z} z^{{1\over 2} + \mu} M\left({1\over 2} + \mu - \kappa, 1 + 2\mu, z\right) $$where M(a,b,z) is Kummer’s solution of the confluent hypergeometric equation.
This can also be expressed by the series (DLMF 13.14.E6): $$ M_{\kappa,\mu}(z) = e^{-{1\over 2} z} z^{{1\over 2} + \mu} \sum_{s=0}^{\infty} {\left({1\over 2} + \mu - \kappa\right)_s \over (1 + 2\mu)_s s!} z^s $$
Whittaker W function (A&S eqn 13.1.33): $$ W_{\kappa,\mu}(z) = e^{-{1\over 2}z} z^{{1\over 2} + \mu} U\left({1\over 2} + \mu - \kappa, 1+2\mu,z\right) $$
where U(a,b,z) is Kummer’s second solution of the confluent hypergeometric equation.
The \(_{p}F_{q}(a_1,a_2,...,a_p;b_1,b_2,...,b_q;z)\) hypergeometric function, where a a list of length p and b a list of length q.
The hypergeometric function. Unlike Maxima’s %f hypergeometric
function, the function hypergeometric is a simplifying
function; also, hypergeometric supports complex double and
big floating point evaluation. For the Gauss hypergeometric function,
that is p = 2 and q = 1, floating point evaluation
outside the unit circle is supported, but in general, it is not
supported.
When the option variable expand_hypergeometric is true (default
is false) and one of the arguments a1 through ap is a
negative integer (a polynomial case), hypergeometric returns an
expanded polynomial.
Examples:
(%i1) hypergeometric([],[],x); (%o1) %e^x
Polynomial cases automatically expand when expand_hypergeometric is true:
(%i2) hypergeometric([-3],[7],x); (%o2) hypergeometric([-3],[7],x) (%i3) hypergeometric([-3],[7],x), expand_hypergeometric : true; (%o3) -x^3/504+3*x^2/56-3*x/7+1
Both double float and big float evaluation is supported:
(%i4) hypergeometric([5.1],[7.1 + %i],0.42);
(%o4) 1.346250786375334 - 0.0559061414208204 %i
(%i5) hypergeometric([5,6],[8], 5.7 - %i);
(%o5) .007375824009774946 - .001049813688578674 %i
(%i6) hypergeometric([5,6],[8], 5.7b0 - %i), fpprec : 30;
(%o6) 7.37582400977494674506442010824b-3
- 1.04981368857867315858055393376b-3 %i
hypergeometric_simp simplifies hypergeometric functions
by applying hgfred
to the arguments of any hypergeometric functions in the expression e.
Only instances of hypergeometric are affected;
any %f, %w, and %m in the expression e are not affected.
Any unsimplified hypergeometric functions are returned unchanged
(instead of changing to %f as hgfred would).
load("hypergeometric"); loads this function.
See also hgfred.
Examples:
(%i1) load ("hypergeometric") $
(%i2) foo : [hypergeometric([1,1], [2], z), hypergeometric([1/2], [1], z)];
(%o2) [hypergeometric([1, 1], [2], z),
1
hypergeometric([-], [1], z)]
2
(%i3) hypergeometric_simp (foo);
log(1 - z) z z/2
(%o3) [- ----------, bessel_i(0, -) %e ]
z 2
(%i4) bar : hypergeometric([n], [m], z + 1);
(%o4) hypergeometric([n], [m], z + 1)
(%i5) hypergeometric_simp (bar);
(%o5) hypergeometric([n], [m], z + 1)
Simplify the generalized hypergeometric function in terms of other, simpler, forms. a is a list of numerator parameters and b is a list of the denominator parameters.
If hgfred cannot simplify the hypergeometric function, it returns
an expression of the form %f[p,q]([a], [b], x) where p is
the number of elements in a, and q is the number of elements
in b. This is the usual
\(_pF_q\)
generalized hypergeometric
function.
(%i1) assume(not(equal(z,0)));
(%o1) [notequal(z, 0)]
(%i2) hgfred([v+1/2],[2*v+1],2*%i*z);
v/2 %i z
4 bessel_j(v, z) gamma(v + 1) %e
(%o2) ---------------------------------------
v
z
(%i3) hgfred([1,1],[2],z);
log(1 - z)
(%o3) - ----------
z
(%i4) hgfred([a,a+1/2],[3/2],z^2);
1 - 2 a 1 - 2 a
(z + 1) - (1 - z)
(%o4) -------------------------------
2 (1 - 2 a) z
It can be beneficial to load orthopoly too as the following example shows. Note that L is the generalized Laguerre polynomial.
(%i5) load("orthopoly")$
(%i6) hgfred([-2],[a],z);
(a - 1)
2 L (z)
2
(%o6) -------------
a (a + 1)
(%i7) ev(%);
2
z 2 z
(%o7) --------- - --- + 1
a (a + 1) a
Next: Functions and Variables for Special Functions, Previous: Hypergeometric Functions, Up: Special Functions [Contents][Index]
The Parabolic Cylinder Functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 19.
Maxima has very limited knowledge of these functions. They
can be returned from function specint.
The parabolic cylinder function parabolic_cylinder_d(v,z). (A&S eqn 19.3.1).
The solution of the Weber differential equation $$ y''(z) + \left(\nu + {1\over 2} - {1\over 4} z^2\right) y(z) = 0 $$
has two independent solutions, one of which is \(D_{\nu}(z),\) the parabolic cylinder d function.
Function specint can return expressions containing
parabolic_cylinder_d(v,z) if the option variable
prefer_d is true.
Previous: Parabolic Cylinder Functions, Up: Special Functions [Contents][Index]
The principal branch of Lambert’s W function W(z) (DLMF 4.13), the solution of $$ z = W(z)e^{W(z)} $$
The k-th branch of Lambert’s W function W(z) (DLMF 4.13), the solution of \(z=W(z)e^{W(z)}.\)
The principal branch, denoted
\(W_p(z)\)
in DLMF, is lambert_w(z) = generalized_lambert_w(0,z).
The other branch with real values, denoted
\(W_m(z)\)
in DLMF, is generalized_lambert_w(-1,z).
The Bateman k function
$$ k_v(x) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \cos(x \tan\theta-v\theta)d\theta $$It is a special case of the confluent hypergeometric function. Maxima can
calculate the Laplace transform of kbateman using laplace
or specint, but has no other knowledge of this function.
The Plasma Dispersion Function $$ {\rm nzeta}(z) = i\sqrt{\pi}e^{-z^2}(1-{\rm erf}(-iz)) $$
Returns realpart(nzeta(z)).
Returns imagpart(nzeta(z)).
Next: Elliptic Functions, Previous: Polynomials [Contents][Index]