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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
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
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
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,
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:
This can be defined by the series
where
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
When \(\mu\pm \nu\) ) is an odd negative integer, the limit must be used.
Next: Airy Functions, Previous: Introduction to Special Functions, Up: Special Functions [Contents][Index]