25 How does complex arithmetic work?
Description
This article is from the sci.fractals FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous contributions by others.
25 How does complex arithmetic work?
It works mostly like regular algebra with a couple additional
formulas:
(note: a, b are reals, "x", "y" are complex, "i" is the square root of
-1)
Powers of "i":
"i"^2 = -1
Addition:
(a+"i"*b)+(c+"i"*d) = (a+c)+"i"*(b+d)
Multiplication:
(a+"i"*b)*(c+"i"*d) = a*c-b*d + "i"*(a*d+b*c)
Division:
(a+"i"*b) / (c+"i"*d) = (a+"i"*b)*(c-"i"*d) / (c^2+d^2)
Exponentiation:
exp(a+"i"*b) = exp(a)*(cos(b)+"i"*sin(b))
Sine:
sin("x") = (exp("i"*"x") - exp(-"i"*"x")) / (2*"i")
Cosine:
cos("x") = (exp("i"*"x") + exp(-"i"*"x")) / 2
Magnitude:
|a+"i"*b| = sqrt(a^2+b^2)
Log:
log(a+"i"*b) = log(|a+"i"*b|)+"i"*arctan(b / a) (Note: log is
multivalued.)
Log (polar coordinates):
log(r e^("i"*a)) = log(r)+"i"*a
Complex powers:
"x"^y = exp(y*log(x))
de Moivre's theorem:
"x"^n = r^n [cos(n*a) + "i"*sin(n*a)] (where n is an integer)
More details can be found in any complex analysis book.
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