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Compute the Laplace transform of expr with respect to the variable t. The integrand expr may contain special functions. The parameter s maybe be named something else; it is determined automatically, as the examples below show where p is used in some places.
The following special functions are handled by specint: incomplete gamma
function, error functions (but not the error function erfi, it is easy to
transform erfi e.g. to the error function erf), exponential
integrals, bessel functions (including products of bessel functions), hankel
functions, hermite and the laguerre polynomials.
Furthermore, specint can handle the hypergeometric function
%f[p,q]([],[],z), the Whittaker function of the first kind
%m[u,k](z) and of the second kind %w[u,k](z).
The result may be in terms of special functions and can include unsimplified hypergeometric functions.
When laplace fails to find a Laplace transform, specint is called.
Because laplace knows more general rules for Laplace transforms, it is
preferable to use laplace and not specint.
demo("hypgeo") displays several examples of Laplace transforms computed by
specint.
Examples:
(%i1) assume (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
Examples for exponential integrals:
(%i4) assume(s>0,a>0,s-a>0)$
(%i5) ratsimp(specint(%e^(a*t)
*(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
log(s)
(%o5) ------
s - a
(%i6) logarc:true$
(%i7) gamma_expand:true$
radcan(specint((cos(t)*expintegral_si(t)
-sin(t)*expintegral_ci(t))*%e^(-s*t),t));
log(s)
(%o8) ------
2
s + 1
ratsimp(specint((2*t*log(a)+2/a*sin(a*t)
-2*t*expintegral_ci(a*t))*%e^(-s*t),t));
2 2
log(s + a )
(%o9) ------------
2
s
Results when using the expansion of gamma_incomplete and when changing
the representation to expintegral_e1:
(%i10) assume(s>0)$
(%i11) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
1
gamma_incomplete(-, k s)
2
(%o11) ------------------------
sqrt(%pi) sqrt(s)
(%i12) gamma_expand:true$
(%i13) specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
erfc(sqrt(k) sqrt(s))
(%o13) ---------------------
sqrt(s)
(%i14) expintrep:expintegral_e1$
(%i15) ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));
a s
a s %e expintegral_e1(a s) - 1
(%o15) - ---------------------------------
a
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 pFq 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
The principal branch of Lambert’s W function W(z) (DLMF 4.13), the solution of
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
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
Returns realpart(nzeta(z)).
Returns imagpart(nzeta(z)).
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