Provisionally complete Complex

[joshuabowers]

Complex implements an imprecise interpretation of the mathematical concept of a field, specifically for representing complex numbers. Currently, it defines the representation of a complex number---i.e. a pair of reals, (a, b) which represent the real and imaginary parts of the prototypical binomial a + bi---alongside a series of mathematical operations which may be calculated upon that field. (This is in keeping with it being a sub-class of the abstract Field; Real is a sibling sub-class which implements a similar structure for reals.)

Currently, Complex does not yet define any implementation for the following sets of functions:

• trigonometric
• arcus
• factorial
• exponential

The first two will require the implementation---on Field---of the hyperbolic functions. Factorial will likely need to be represented as the gamma function, which currently makes my head hurt. Exponentiation can be implemented in terms of the trigonometric functions via De Moivre's formula.

It should be noted that the current implementation of the complex logarithm, and some of the likely solutions suggested above, yield only the principal value of the applied function, ignoring the infinite generalized unbounded solutions, by binding the result to a particular domain. For the logarithm, this constrains the resultant imaginary part to lie within (-pi, pi]. A more general approach---which takes into consideration the branch cuts would likely be desirable, but would require Complex to likely define its component parts in terms of Field, with a new sub-class representing variables introduced; conversely, the return type of Field operations would need to be altered to allow for AST node-structures to be allowed, which would allow methods like logarithm to return new sub-trees.

[joshuabowers]

• The principle values of the inverse hyperbolics should be calculable via the complex logarithm.
• The complex hyperbolics can likely be found in terms of multiply and the complex trigonometric functions. E.g. cosh(z) === cos(i * z), where z is complex, the argument i * z is the complex multiplication of i and z, and cos is the complex implementation.
• The complex arcus functions can be implemented in terms of complex logarithms and complex exponentiation. (I.e., the square root is the same as raising a value to a power of 0.5; so the square root of a complex would be equivalent to z ** 0.5, or using the Field API: z.raise(new Complex(0.5, 0)) for z complex.)

[joshuabowers]

A potential approach to implementing the Gamma function: using the Lanczos Approximation. The python example seems relatively simple to translate, and should be functional on complex numbers.

A few notes about implementing Lanczos:

• Precision of the approximated result is governed by (seemingly) arbitrarily chosen multiplicative constants, which also govern the set of coefficients, p, which each term in the convergent series get multiplied by. Much of this can be pre-computed---indeed, the python source linked above already has those values listed for a given g. Other sets of coefficients can, for example, be found here
• The gamma function isn't just about defining a factorial analog for complex numbers---which was the primary concern; rather, it extends the concept of the factorial to most of the reals, modulo the non-positive integers. This might imply a general use case which would be desirable to abstract into Field so that both Real and Complex can utilize the functionality. This might also potentially result in a reimplmentation of Real.prototype.factorial, so that it would be applicable to the broader set of numbers.
• Lanczos typically seems to be defined on an argument of z + 1, which might be weird to represent.
• Also common for approximations to define themselves in terms of a logarithm of the gamma function, rather than the full gamma, as it tends to grow exponentially.

[joshuabowers]

Done, as of aeada1c3b31762e6639595e537711def26fbe21b. While further refinement is likely warranted at some point, Complex numbers are firmly integrated into the app.