Terms with complex numbers can be calculated here.
Possible calculations are addition, subtraction, division, multiplication and exponentiation.
In addition, some functions can also be used in the terms.
These functions all also support complex function arguments.
Supported functions: exp(z), cos(z), sin(z), cosh(z), sinh(z), log(z), abs(z), arg(z), conjg(z), real(z), imag(z) with z ∈ ℂ
Supported literals: i, e, PI
Representation
Input
Result
Representation options for complex numbers
There are essentially 3 different representations:
Arithmetic representation: z = a + i b
Here a is the real part of z and b is the imaginary part of z.
Exponential representation: z = |z| e^{iφ}
Here |z| the amount of z and φ is the argument of z.
The following applies:
|z|^{2} = a^{2} + b^{2}
a = |z| cos φ
b = |z| sin φ
This results in the third possible representation.
Trigonometric representation: z = |z| (cos φ + i sin φ)
This representation is often referred to as the polar form.
In this representation it is obvious that
φ is the polar coordinate angle of the complex number z in the complex plane and
|z| is the distance of the complex number to the origin.
Complex arithmetic operations
For the two complex numbers
z_{1} = a_{1} + i b_{1} = |z_{1}| e^{iφ1} und
z_{2} = a_{2} + i b_{2} = |z_{2}| e^{iφ2}
applies:
z_{1} + z_{2} = (a_{1} + a_{2}) + i (b_{1} + b_{2})
z_{1} - z_{2} = (a_{1} - a_{2}) + i (b_{1} - b_{2})
z_{1} · z_{2} = (a_{1}a_{2} - b_{1}b_{2}) + i (a_{1}b_{2} + a_{2}b_{1}) = |z_{1}| |z_{2}| e^{i(φ1+φ2)}
z_{1} / z_{2} = ((a_{1}a_{2} + b_{1}b_{2}) + i ( a_{2}b_{1} - a_{1}b_{2}) ) / (a_{2}^{2} + b_{2}^{2}) = |z_{1}|/|z_{2}| e^{i(φ1-φ2)}
The following applies to raising the complex number z to the power of a real exponent c:
z^{c} = (|z| e^{iφ})^{c} = |z|^{c} e^{iφc}
The following applies to amounts of product and quotient:
|z_{1} · z_{2}| = |z_{1}| · |z_{2}|
|z_{1} / z_{2}| = |z_{1}| / |z_{2}|
Elementary functions with complex argument
For z ∈ ℂ with z = x + i y the following relationships apply:
e^{z} = e^{x} (cos y + i sin y)
sin z = sin x cosh y + i cos x sinh y
cos z = cos x cosh y - i sin x sinh y
sinh z = sinh x cos y + i cosh x sin y
cosh z = cosh x cos y + i sinh x sin y
log z = ln|z| + i arg(z)
z^{c} = e^{c log(z)}, c ∈ ℂ