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16.1 Introduction to Special Functions | ||
16.2 Functions and Variables for Special Functions |
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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 %he[n] (z) Hermite polynomial (Nota bene:he
, noth
. See A&S 22.5.18) %p[u,v] (z) Legendre function %q[u,v] (z) Legendre function, 2nd kind hstruve[n] (z) Struve H function lstruve[n] (z) Struve L function %f[p,q] ([], [], expr) Generalized Hypergeometric function gamma() Gamma function gammagreek(a,z) Incomplete gamma function gammaincomplete(a,z) Tail of incomplete gamma function slommel %m[u,k] (z) Whittaker function, 1st kind %w[u,k] (z) Whittaker function, 2nd kind erfc (z) Complement of the erf function ei (z) Exponential integral (?) kelliptic (z) Complete elliptic integral of the first kind (K) %d [n] (z) Parabolic cylinder function
Categories: Bessel functions · Airy functions · Special functions
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The Airy function Ai, as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, Section 10.4.
The Airy equation diff (y(x), x, 2) - x y(x) = 0
has two
linearly independent solutions, y = Ai(x)
and y = Bi(x)
.
The derivative diff (airy_ai(x), x)
is airy_dai(x)
.
If the argument x
is a real or complex floating point
number, the numerical value of airy_ai
is returned
when possible.
See also airy_bi
, airy_dai
, airy_dbi
.
Categories: Airy functions
The derivative of the Airy function Ai airy_ai(x)
.
See airy_ai
.
Categories: Airy functions
The Airy function Bi, as defined in Abramowitz and Stegun,
Handbook of Mathematical Functions, Section 10.4,
is the second solution of the Airy equation
diff (y(x), x, 2) - x y(x) = 0
.
If the argument x
is a real or complex floating point number,
the numerical value of airy_bi
is returned when possible.
In other cases the unevaluated expression is returned.
The derivative diff (airy_bi(x), x)
is airy_dbi(x)
.
See airy_ai
, airy_dbi
.
Categories: Airy functions
The derivative of the Airy Bi function airy_bi(x)
.
See airy_ai
and airy_bi
.
Categories: Airy functions
asympa
is a package for asymptotic analysis. The package contains
simplification functions for asymptotic analysis, including the "big O"
and "little o" functions that are widely used in complexity analysis and
numerical analysis.
load ("asympa")
loads this package.
Categories: Package asympa
The Bessel function of the first kind.
This function is deprecated. Write bessel_j (z, a)
instead.
The Bessel function of the first kind of order v and argument z.
bessel_j
computes the array besselarray
such that
besselarray [i] = bessel_j [i + v - int(v)] (z)
for i
from zero to int(v)
.
bessel_j
is defined as
inf ==== k - v - 2 k v + 2 k \ (- 1) 2 z > -------------------------- / k! gamma(v + k + 1) ==== k = 0
although the infinite series is not used for computations.
Categories: Bessel functions
The Bessel function of the second kind of order v and argument z.
bessel_y
computes the array besselarray
such that
besselarray [i] = bessel_y [i + v - int(v)] (z)
for i
from zero to int(v)
.
bessel_y
is defined as
cos(%pi v) bessel_j(v, z) - bessel_j(-v, z) ------------------------------------------- sin(%pi v)
when v is not an integer. When v is an integer n, the limit as v approaches n is taken.
Categories: Bessel functions
The modified Bessel function of the first kind of order v and argument z.
bessel_i
computes the array besselarray
such that
besselarray [i] = bessel_i [i + v - int(v)] (z)
for i
from zero to int(v)
.
bessel_i
is defined as
inf ==== - v - 2 k v + 2 k \ 2 z > ------------------- / k! gamma(v + k + 1) ==== k = 0
although the infinite series is not used for computations.
Categories: Bessel functions
The modified Bessel function of the second kind of order v and argument z.
bessel_k
computes the array besselarray
such that
besselarray [i] = bessel_k [i + v - int(v)] (z)
for i
from zero to int(v)
.
bessel_k
is defined as
%pi csc(%pi v) (bessel_i(-v, z) - bessel_i(v, z)) ------------------------------------------------- 2
when v is not an integer. If v is an integer n, then the limit as v approaches n is taken.
Categories: Bessel functions
Default value: false
Controls expansion of the Bessel functions when the order is half of
an odd integer. In this case, the Bessel 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); 2 z sin(z) cos(z) (%o4) sqrt(---) (------ - ------) %pi 2 z z
Categories: Bessel functions · Simplification flags and variables
The scaled modified Bessel function of the first kind of order
v and argument z. That is, scaled_bessel_i(v,z) =
exp(-abs(z))*bessel_i(v, z). This function is particularly useful
for calculating bessel_i 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
Identical to scaled_bessel_i(0,z)
.
Categories: Bessel functions
Identical to scaled_bessel_i(1,z)
.
The beta function, defined as gamma(x) gamma(y)/gamma(x + y)
.
Categories: Gamma and factorial functions
The gamma function.
See also makegamma
.
The variable gammalim
controls simplification of the gamma function.
The Euler-Mascheroni constant is %gamma
.
Categories: Gamma and factorial functions
Default value: 1000000
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.
Converts a into a Poisson encoding.
Categories: Poisson series
Transforms instances of binomial, gamma, and beta functions in expr into factorials.
See also makegamma
.
Categories: Gamma and factorial functions
Transforms instances of binomial, factorial, and beta functions in expr into gamma functions.
See also makefact
.
Categories: Gamma and factorial functions
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
Converts a from Poisson encoding to general
representation. If a is not in Poisson form, outofpois
carries out the conversion,
i.e., the return value is outofpois (intopois (a))
.
This function is thus a canonical simplifier
for sums of powers of sine and cosine terms of a particular type.
Categories: Poisson series
Differentiates a with respect to b. b must occur only in the trig arguments or only in the coefficients.
Categories: Poisson series
Functionally identical to intopois (a^b)
.
b must be a positive integer.
Categories: Poisson series
Integrates in a similarly restricted sense (to
poisdiff
). Non-periodic terms in b are dropped if b is in the trig
arguments.
Categories: Poisson series
Default value: 5
poislim
determines the domain of the coefficients in
the arguments of the trig functions. The initial value of 5
corresponds to the interval [-2^(5-1)+1,2^(5-1)], or [-15,16], but it
can be set to [-2^(n-1)+1, 2^(n-1)].
Categories: Poisson series
will map the functions sinfn on the sine terms and cosfn on the cosine terms of the Poisson series given. sinfn and cosfn are functions of two arguments which are a coefficient and a trigonometric part of a term in series respectively.
Categories: Poisson series
Is functionally identical to intopois (a + b)
.
Categories: Poisson series
Converts a into a Poisson series for a in general representation.
Categories: Poisson series
The symbol /P/
follows the line label of Poisson series
expressions.
Categories: Poisson series
Substitutes a for b in c. c is a Poisson series.
(1) Where B is a variable u, v, w, x, y, or z,
then a must be an
expression linear in those variables (e.g., 6*u + 4*v
).
(2) Where b is other than those variables, then a must also be free of those variables, and furthermore, free of sines or cosines.
poissubst (a, b, c, d, n)
is a special type of substitution which
operates on a and b as in type (1) above, but where d is a Poisson
series, expands cos(d)
and sin(d)
to order n so as to provide the
result of substituting a + d
for b in c. The idea is that d is an
expansion in terms of a small parameter. For example,
poissubst (u, v, cos(v), %e, 3)
yields cos(u)*(1 - %e^2/2) - sin(u)*(%e - %e^3/6)
.
Categories: Poisson series
Is functionally identical to intopois (a*b)
.
Categories: Poisson series
is a reserved function name which (if the user has defined
it) gets applied during Poisson multiplication. It is a predicate
function of 6 arguments which are the coefficients of the u, v, ..., z
in a term. Terms for which poistrim
is true
(for the coefficients of
that term) are eliminated during multiplication.
Categories: Poisson series
Prints a Poisson series in a readable format. In common
with outofpois
, it will convert a into a Poisson encoding first, if
necessary.
Categories: Poisson series · Display functions
The derivative of log (gamma (x))
of order n+1
.
Thus, psi[0](x)
is the first derivative,
psi[1](x)
is the second derivative, etc.
Maxima does not know how, in general, to compute a numerical value of
psi
, but it can compute some exact values for rational args.
Several variables control what range of rational args psi
will
return an exact value, if possible. See maxpsiposint
,
maxpsinegint
, maxpsifracnum
, and maxpsifracnum
.
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
will return an exact value.
The function bfpsi
in the bffac
package can compute
numerical values.
Categories: Gamma and factorial functions
Default value: 20
maxpsiposint
is the largest positive value for which
psi[n](x)
will try to compute an exact value.
Categories: Gamma and factorial functions
Default value: -10
maxpsinegint
is the most negative value for which
psi[n](x)
will try to compute an exact value. That is if
x is less than maxnegint
, psi[n](x)
will not
return simplified answer, even if it could.
Categories: Gamma and factorial functions
Default value: 6
Let x be a rational number less than one of the form p/q
.
If p
is greater than maxpsifracnum
, then
psi[n](x)
will not try to return a simplified
value.
Categories: Gamma and factorial functions
Default value: 6
Let x be a rational number less than one of the form p/q
.
If q
is greater than maxpsifracdenom
, then
psi[n](x)
will not try to return a simplified
value.
Categories: Gamma and factorial functions
Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions.
If specint
cannot compute the integral, the return value may
contain various Lisp symbols, including
other-defint-to-follow-negtest
,
other-lt-exponential-to-follow
,
product-of-y-with-nofract-indices
, etc.; this is a bug.
demo(hypgeo)
displays several examples of Laplace transforms computed by specint
.
Examples:
(%i1) assume (p > 0, a > 0); (%o1) [p > 0, a > 0] (%i2) specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t); sqrt(%pi) (%o2) ------------ a 3/2 2 (p + -) 4 (%i3) specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2)) * exp(-p*t), t); - a/p sqrt(a) %e (%o3) --------------- 2 p
Categories: Laplace transform
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