I have used taylor polynomials to approximate floating-point values before. I am wondering whether there is a way to still apply them to functions with complex arguments, and if there is a way what its constraints would be. For example, to calculate the sine of a floating-point number to a specific precision I keep adding terms until the error value is less than the precision. However, I noticed (by testing it with the taylor expansion of ln(x) ) that this gives the wrong value when I use complex numbers (with the standard operators appropriately defined). For example:
ln(0.5)~=-(0.5+(0.5^2)/2+(0.5^3)/3) --- gives -0.66, real is -0.69 Close enough ln(0.5i)~=-(0.5i+(0.5i^2)/2+(0.5i^3)/3) --- gives -0.25+0.33i, real is -0.69+1.57i WHAAT?!
How can I approximate imaginary functions to within an error range? Basically I need a method such that if I give you any given function's definition, a complex number, and a rational number (floating-point value) you can give me the value of that given function, at that given complex number, accurate in both real and imaginary parts, to within +- the given rational number. Clearly standard taylor polynomials dont always work, so what does?