Complex Functions

The complex functions are those functions that:

• take one or more arguments that are complex expressions or double expressions and
• return a complex value

• i the square root of -1

The set of available double functions is:

• one argument functions:

  Function Name	Description	Notes
  acsc	inverse cosecant	acsc(z) = asin(1.0 / z)
  acsch	inverse hyperbolic cosecant	acsch(z) = asinh(1.0 / z)
  acos	inverse cosine	acos(z) = -i * log(z + i * sqrt(1.0 - z * z))
  acosh	inverse hyperbolic cosine	acosh(z) = log(z + sqrt(z * z - 1.0))
  acot	inverse cotangent	acot(z) = atan(1.0 / z)
  acoth	inverse hyperbolic cotangent	acoth(z) = atanh(1.0 / z)
  asec	inverse secant	asec(z) = acos(1.0 / z)
  asech	inverse hyperbolic cosecant	asech(z) = acosh(1.0 / z)
  asin	inverse sin	asin(z) = -i * log(i * z + sqrt(1.0 - z * z))
  asinh	inverse hyperbolic sine	asinh(z) = log(z + sqrt(z * z + 1.0))
  atan	inverse tangent	result = tan(z), z = atan(result)
  atanh	inverse hyperbolic tangent	atanh(value) = log((value + 1.0) / (value - 1.0)) / 2.0
  conj	complex conjugate	if z = [real, imaginary] then conj(z) = [real, -imaginary]
  cos	cosine	cos(z) = (exp(i * z) + exp(-i * z)) / 2
  cosh	hyperbolic cosine	cosh(z) = (exp(z) + exp(-z)) / 2.0
  cot	cotangent	cot(z) = cos(z) / sin(z)
  coth	hyperbolic cotangent	coth(z) = cosh(z) / sinh(z)
  csc	cosecant	csc(z) = 1.0 / sin(z)
  csch	hyperbolic cosecant	csch(z) = 1.0 / sinh(z)
  exp	exponential	if z = [real, imaginary] then exp(z) = exp(real) * (cos(imaginary) + i * sin(imaginary))
  log	natural log	for z = [real, imaginary], we can also express z in terms of a distance 'r' and an angle 'theta'. log(z) = log(r) + i * theta. Log(z) is a multivalued function
  sec	secant	sec(z) = 1.0 / cos(z)
  sech	hyperbolic secant	sech(z) = 1.0 / cosh(z)
  sin	sine	sin(z) = (exp(i * z) - exp(-i * z)) / 2.0
  sinh	hyperbolic sine	sinh(z) = (exp(z) - exp(-z)) / 2.0
  sqrt	square root	The square root of z is the number that when multipled by itself is equal to z. For example: sqrt([0, 8]) == [2, 2]
  tan	tangent	tan(z) = sin(z) / cos(z)
  tanh	hyperbolic tangent	tanh(z) = sinh(z) / cosh(z)

• two argument functions:

  Function Name	Description	Notes
  pow	the power function	This function raises the first argument to the power of the second argument Here are some examples pow([2, 2], [2, 0]) = [0, 8] pow([1, 2], [3, 4]) = [0.12901, 0.0339241] This function can be used with a mixture of complex and real types as in: pow([2, 2], 2) pow(2, [123, 456]) NOTE: You can also write pow(z1, z2) as "z1 ^ z2"

• three argument functions:

  Function Name	Description	Notes
  rotate	rotates a point a certain number of degrees around another point	As an example: z = [2, 0]; z = rotate(z, [0, 0], 90); This example says "rotate the point 'z' 90 degrees around the point [0, 0] (the origin)". In this case, the result will be [0, 2]