### besselj =====================================================================

Tags
----
scripting, JS, internal, treenode, general concept

Parameters
----------
x,n

General description
-------------------

Returns a solution to the Bessel differential equation (Bessel function of first kind).

Recurrence relation:

  besselj(x,n) = (2/x) (n-1) besselj(x,n-1) - besselj(x,n-2)

  besselj(x,1/2) = sqrt(2/pi/x) sin(x)

  besselj(x,-1/2) = sqrt(2/pi/x) cos(x)

For negative n, reorder the recurrence relation as:

  besselj(x,n-2) = (2/x) (n-1) besselj(x,n-1) - besselj(x,n)

Substitute n+2 for n to obtain

  besselj(x,n) = (2/x) (n+1) besselj(x,n+1) - besselj(x,n+2)

Examples:

  besselj(x,3/2) = (1/x) besselj(x,1/2) - besselj(x,-1/2)

  besselj(x,-3/2) = -(1/x) besselj(x,-1/2) - besselj(x,1/2)

###



Eval_besselj = ->
  push(cadr(p1))
  Eval()
  push(caddr(p1))
  Eval()
  besselj()

besselj = ->
  save()
  yybesselj()
  restore()

#define X p1
#define N p2
#define SGN p3

yybesselj = ->
  d = 0.0
  n = 0

  p2 = pop()
  p1 = pop()

  push(p2)
  n = pop_integer()

  # numerical result

  if (isdouble(p1) && !isNaN(n))
    d = jn(n, p1.d)
    push_double(d)
    return

  # bessej(0,0) = 1

  if (isZeroAtomOrTensor(p1) && isZeroAtomOrTensor(p2))
    push_integer(1)
    return

  # besselj(0,n) = 0

  if (isZeroAtomOrTensor(p1) && !isNaN(n))
    push_integer(0)
    return

  # half arguments

  if (p2.k == NUM && MEQUAL(p2.q.b, 2))

    # n = 1/2

    if (MEQUAL(p2.q.a, 1))
      if evaluatingAsFloats
        push_double(2.0 / Math.PI)
      else
        push_integer(2)
        push_symbol(PI)
        divide()
      push(p1)
      divide()
      push_rational(1, 2)
      power()
      push(p1)
      sine()
      multiply()
      return

    # n = -1/2

    if (MEQUAL(p2.q.a, -1))
      if evaluatingAsFloats
        push_double(2.0 / Math.PI)
      else
        push_integer(2)
        push_symbol(PI)
        divide()
      push(p1)
      divide()
      push_rational(1, 2)
      power()
      push(p1)
      cosine()
      multiply()
      return

    # besselj(x,n) = (2/x) (n-sgn(n)) besselj(x,n-sgn(n)) - besselj(x,n-2*sgn(n))

    push_integer(MSIGN(p2.q.a))
    p3 = pop()

    push_integer(2)
    push(p1)
    divide()
    push(p2)
    push(p3)
    subtract()
    multiply()
    push(p1)
    push(p2)
    push(p3)
    subtract()
    besselj()
    multiply()
    push(p1)
    push(p2)
    push_integer(2)
    push(p3)
    multiply()
    subtract()
    besselj()
    subtract()

    return

  #if 0 # test cases needed
  if (isnegativeterm(p1))
    push(p1)
    negate()
    push(p2)
    power()
    push(p1)
    push(p2)
    negate()
    power()
    multiply()
    push_symbol(BESSELJ)
    push(p1)
    negate()
    push(p2)
    list(3)
    multiply()
    return

  if (isnegativeterm(p2))
    push_integer(-1)
    push(p2)
    power()
    push_symbol(BESSELJ)
    push(p1)
    push(p2)
    negate()
    list(3)
    multiply()
    return
  #endif

  push(symbol(BESSELJ))
  push(p1)
  push(p2)
  list(3)