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A look-up table for Complex number equations

As you know, complex numbers have a real and an imaginary part. They can be represented as an angle too. You can add, subtract, multiply and divide them but you probably didn't know that you can use them with functions. I can't think why you would want to take the log of a complex number, but you can. Most calculator's are not programmed to handle them or at least, not fully. Have a look at the article on function equations for alternate forms too.
Note that some equations use trigonometry and therefore have an infinite number of solutions due to their cyclic nature.
Complex numbers provide a short-cut to answers for example, in electronics, they are used to show what a current tends to which gives a practical answer as you don't need to know that it doesn't actually get to that number like 1/x (where x>0) never actually gets to 0, just near enough.
While I'm on the subject of electronics, electricity does NOT travel at the speed of light. It cannot because electrons are what electricity is and electrons have mass, therefore they cannot travel at that speed. The only thing that travels at the speed of light (in free space) is light AND gravity (yes, gravity).
Electricity is actually very slow. It can take a few seconds to move one centimetre but its reaction is practically instantaneous for example, a snooker queue can be pushed really slowly but as soon as you stop pushing one end, the other stops like if you disconnect the battery, a light goes out from the other side of a room.

Addition

(a + bi) + (c + di) = (a + c) + i(b + d)

Subtraction

(a + bi) - (c + di) = (a - c) + i(b - d)

Multiplication

(r1∠θ1)(r2∠θ2) = (r1•r2)∠(θ1 + θ2)
(a + bi)(c+di) = (ac - bd) + i(ad + bc)

Division

r1∠θ1  =  r1 ∠ θ1 - θ2
r2∠θ2 r2
a + bi  =  (ac + bd) + i(bc - ad)
c + di c2+d2

Logrithm and exponential

ln(a + bi) = ln|a+bi| + i•arg(a+bi) ← arg() is explained later on
ln(r∠θ) = ln(r) + θi (the result is in complex number form)
ea+bi = ea(cos(b) + i•sin(b)) = ea∠b
note that an expression that's something like cos(θ) + i•sin(θ) = 1∠θ
Also ln has a cyclic part so:
ln((-i)2) ≠ 2•ln(-i)
ln((-i)2) = πi
2•ln(-i) = -πi

Powers

(r∠θ)n = rn∠nθ

(a+bi)n = |a + bi|n•(cos(nθ) + i•sin(nθ)) if θ = arg(a+bi)

(a+bi)(c+di) = e(c+di)•ln(a+bi)

(a+bi)(c+di) = (cos(f) + i•sin(f))ec•ln|a+bi| - d•arg(a+bi) if f=c•arg(a + bi) + d•ln|a + bi|

Trigonometry

sin(a + bi) = sin(a)•cosh(b) + i•cos(a)•sinh(b)

cos(a + bi) = cos(a)•cosh(b) - i•sin(a)•sinh(b)

tan(a+bi) =  tan(a) + tanh(b)•i  =  sin(a)•cos(a) + i•sinh(b)•cosh(b)
1 - i•tan(a)•tanh(b) cos2(a) + cosh2(b) - 1
The above can be further changed because sin(a)•cos(a) = 0.5•sin(2a) and cos2(a) = 0.5(1+cos(2a)) etc.
if z = a + bi then
sin-1(z) = -i•ln(iz+√(1-z2))
cos-1(z) = i•ln(iz+√(1-z2)) +  π
2
tan-1(z) =  i (ln(1-iz) - ln(1+iz))
2

Hyperbolia

See my article on function equations for these which work for complex numbers

Arg(x + yi) - this gives the angle θ when the complex number is in polar form

x y arg(x + yi)
>0 ≠0 tan-1(y/x)
>0 =0 0
0 >0 90o or π / 2
0 <0 -90o or -π / 2
<0 >0 tan-1(y/x) + 180o (or π)
<0 <0 tan-1(y/x) - 180o (or -π)
<0 0 180o (or π)
0 0 does not exist
Diagram of the arg() function



Have you found an error or do you want to add more information to these pages?
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