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25 How does complex arithmetic work?




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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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