Complex Numbers for FreePascal

Author: Paul Lipkowski

About

I decided to write my own complex numbers mechanism, as FreePascal's built-in mechanism isn't sufficient enough for me.

Features

As of now

• basic routines, type casting
• powers, square and cubic roots
• exponential and logarithmic functions
• trigonometric functions and their inverses
• hyperbolic and area functions
• MinusOneTo, ImagTo, MinusImagTo
• isZero, isNatural, isInteger, isReal, isNotReal, isImaginary, isComplex boolean functions
• ComplexNumPolar (initializing a complex number with polar coords)
• gamma, gammaln functions
• lower and upper incomplete gamma functions
• erf, erfc, erfi functions
• beta, incomplete beta, regularized incomplete beta functions
• E_n function (exponential integral) with branches
• Ei function
• li (logarithm integral)
• sinc function
• Si, Ci functions (sine integral, cosine integral)
• Fresnel integrals
• Newton symbol (z,n)
• Dirichlet eta function
• Riemann's zeta function
• Bernoulli numbers
• Lambert's W function with all integer branches
• Infinite power tower h(z) = z^z^z^...
• Infinities
• Complex round functions

Issues

• The complex type substitutes Extended type almost plug and play, except for:
  • inequalities (for mathematical reasons one can't set any linear order for complex numbers)
  • putting them into write/writeln - they must be cast with AnsiString().
• Bernoulli numbers generation may produce wrong results at n >= 40.

References

1. W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery; 'Numerical Recipes. The Art of Scientific Computing. Third Edition'; Cambridge University Press (2007)
2. L. Lorentzen, H. Waadeland; 'Continued fractions with applications'; North-Holland Publishing Co. (1992)
3. D. Harvey; 'A multimodular algorithm for computing Bernoulli numbers'; Math. Comput., 79 (272): 2361–2370 (2010)
4. L. Lóczi; 'Guaranteed- and high-precision evaluation of the Lambert W function'. Appl. Math. Comput., 433: 127406 (2022)
5. github.com/IstvanMezo/LambertW-function