icon-komplexe-Zahl Complex Numbers

Calculator

Graphical

Complex functions

Complex numbers general

The complex numbers are an extension of the real numbers. Many computational rules of real numbers can be applied to complex numbers.

The theory of analytic functions dealt with functions of a complex variable.

The Origin of complex numbers is due to the release of algebraic equations. The origin of the theory of imaginary numbers, that is, all numbers whose square is a negative real number, going to the Italian mathematician Gerolamo Cardano and Rafael Bombelli in the 16th Century. The introduction of the imaginary unit i as the new number is attributed to Leonhard Euler.

Gauss plane

The complex numbers are two-dimensional and can be used as vectors in the Gaussian plane of numbers represent. On the horizontal axis (Re) of the real part and on the vertical axis is applied (Im) of the imaginary part of the complex number. Vectors may also be similar to either the complex number in Cartesian coordinates (x, y) or polar coordinates (r, φ) can be expressed.

Gaußsche-Zahlenebene

Calculator (Cartesian)

z = x+iy

x= y=

Calculator (Polar)

z=rcosφ+isinφ

r= φ=

Relationships of complex conjugate numbers

For complex conjugate numbers, the following relations hold.

z = z wenn z reell is;

z1+z2 = z1+z2

z1z2 = z1z2

z1z2 = z1z2

zz = x2+y2

Definitions and notations for complex numbers

A complex number z consiss of a real part x and an imaginary part y. The imaginary part is characterized by the imaginary unit i.

z = x+iy

mitx,yRandi=-1

The complex conjugate to z consiss of a real part x and the negative imaginary part y. This corresponds to a reflection in the real axis in the Gaussian plane.

z = x-iy

The amount of a complex number corresponds in the Gaussian plane with the length of the vector.

z = x2+y2

The complex number in polar coordinates.

z = rcosφ+isinφ = reiφ

withr=z=x2+y2

andφ=atanyx

and the Euler Formulacosφ+isinφ=eiφ

Rules for computing with complex numbers

Addition and subtraction of complex numbers

The addition and subtraction of complex numbers corresponding to the addition and subtraction of the position vectors. That the real and imaginary components are added or subtracted.

With z1 = x1+iy1 and z2 = x2+iy2 is

z1+z2 = x1+x2 +i y1+y2

z1-z2 = x1-x2 +i y1-y2

Multiplication of complex numbers

The multiplication is done by the brackets, taking into account the relationship i 2 = -1 be multiplied out.

With z1 = x1+iy1 and z2 = x2+iy2 is

z1z2 = x1+iy1 x2+iy2 = x1x2-y1y2 +i x1y2+y1x2

The multiplication of complex numbers can also be done in trigonometric or exponential form.

Withz1 = r1cosφ+isinφ = r1eiφ

andz2 = r2cosψ+isinψ = r2eiψis

z1z2 = r1r2cosφ+ψ+isinφ+ψ = r1r2eiφ+ψ

Calculator: multiplication of complex numbers

z1 = x1+iy1

z2 = x2+iy2

x1= + i y1=

x2= + i y2=

Division of complex numbers

The division is carried out by the fraction is expanded with the complex conjugate of the denominator.

With z1 = x1+iy1 and z2 = x2+iy2 is

z1z2 = x1+iy1 x2+iy2 = x1+iy1 x2+iy2 x2-iy2 x2-iy2 = x1x2+y1y2 x22+y22 +i x2y1-x1y2 x22+y22

The division of complex numbers can also be done in trigonometric or exponential form.

Withz1 = r1cosφ+isinφ = r1eiφ

andz2 = r2cosψ+isinψ = r2eiψis

z1z2 = r1r2cosφ-ψ+isinφ-ψ = r1r2eiφ-ψ

Calculator: Division of complex numbers

z1 = x1+iy1

z2 = x2+iy2

x1= + i y1=

x2= + i y2=

Basic complex functions f(z)

Complex number cartesian

z=x+iy

Real part

Re(z)=x

Imaginary part

Im(z)=y

Konjugated complex number

z=x-iy

Amount

z=x2+y2

Argument

arg(z)=φ=atanyx

Polar

z=zcosφ+isinφ

Quadratic

z2=x2-y2+ixy

Inverse

1z=xx2+y2+i-yx2+y2

Quadratic inverse

1z2=x2-y2x2+y22+i-2xyx2+y22

Root

z=±x+x2+y22±i-x+x2+y22

Exponential function

ez=excosy+iexsiny

Logarithm

lnz=12lnx2+y2+iatanyx

Sinus

sinz=sinxcoshy+icosxsinhy

Cosinus

cosz=cosxcoshy-isinxsinhy

Sinus Hyperbolicus

sinhz=sinhxcosy+icoshxsiny

Cosinus Hyperbolicus

coshz=coshxcosy-isinhxsiny

Tangens

tanz=sin2xcos2x+cosh2y+isinh2ycos2x+cosh2y

Powers of complex numbers

The rose of a complex number in the n-th natural potency is done by using the formula of Moivre.

zn = rncosnφ+isinnφ = rneinφ

withr=z=x2+y2

andφ=atanyx

andi0=1,i1=i,i2=-1,i3=-i...

andnN

or with the binomial theorem

zn =x+iyn

= k = 0, k gerade n n k -1 k 2 x n - k y k

+i k = 1, k ungerade n n k -1 k-1 2 x n - k y k

Calculator: Complex Binomial Theorem

n=

For general complex exponents is:

zω = eωlnz

with zω as main value. If ω is not rational so there are infinite number of solutions.

power-complex-numbers