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The Maxima on-line user's manual

Algebra Calculator

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Hankel_1 Calculator

Hankel_1

Function: hankel_1 (<v>, <z>) The Hankel function of the first kind of order v and argument z (A&S 9.1.3). hankel_1 is defined as

bessel_j(v,z) + %i * bessel_y(v,z)

Maxima evaluates hankel_1 numerically for a real order v and complex argument z in float precision. The numerical evaluation in bigfloat precision and for a complex order v 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)              .2422684576748738 - 1.471472392670243 %i
          (%i2) hankel_1(1,0.5+%i);
          (%o2)             - .2558287994862166 %i - 0.239575601883016

A complex order v is not supported. Maxima returns a noun form:

          (%i3) hankel_1(%i,0.5+%i);
          (%o3)                       hankel_1(%i, %i + 0.5)

Expansion of hankel_1 when besselexpand is true:

          (%i4) hankel_1(1/2,z),besselexpand:true;
                                sqrt(2) sin(z) - sqrt(2) %i cos(z)
          (%o4)                 ----------------------------------
                                        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:

          (%i5) diff(hankel_1(v,z),z);
                              hankel_1(v - 1, z) - hankel_1(v + 1, z)
          (%o5)               ---------------------------------------
                                                 2
          (%i6) diff(hankel_1(v,z),v);
                                       d
          (%o6)                        -- (hankel_1(v, z))
                                       dv

(%o1)                                true
(%i2) 

Hankel_1 Example

Related Examples