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The Maxima on-line user's manual

Algebra Calculator

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Specint Calculator

Specint

Function: specint (exp(- s*<t>) * <expr>, <t>) Compute the Laplace transform of <expr> with respect to the variable <t>. The integrand <expr> may contain special functions.

assume (p > 0, a > 0);
 specint (t^(1/2) * exp(-a*t/4) * exp(-p*t), t);
 specint (t^(1/2) * bessel_j(1, 2 * a^(1/2) * t^(1/2))                        * exp(-p*t), t);
 assume(s>0,a>0,s-a>0);
 ratsimp(specint(%e^(a*t)*(log(a)+expintegral_e1(a*t))*%e^(-s*t),t));
 logarc:true;
 gamma_expand:true;
 radcan(specint((cos(t)*expintegral_si(t)                               -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
 ratsimp(specint((2*t*log(a)+2/a*sin(a*t)                                -2*t*expintegral_ci(a*t))*%e^(-s*t),t));
 assume(s>0);
 specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
 gamma_expand:true;
 specint(1/sqrt(%pi*t)*unit_step(t-k)*%e^(-s*t),t);
 expintrep:expintegral_e1;
 ratsimp(specint(1/(t+a)^2*%e^(-s*t),t));

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 hypergeomtric 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$
          (%i8) radcan(specint((cos(t)*expintegral_si(t)
                               -sin(t)*expintegral_ci(t))*%e^(-s*t),t));
                                              log(s)
          (%o8)                               ------
                                               2
                                              s  + 1

          (%i9) 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

(%o1)                                true
(%i2) 

Specint Example

Related Examples