This online Complex Number Functions Calculator computes some functions of a complex number (variable). When typing the imaginary part of a complex number in the appropriate field of the calculator, make sure that the symbol ‘i‘, representing the imaginary unit, is adjacent to the numeric part without space.
Functions of one complex variable
Our complex number functions calculator computes the following functions of a complex variable \(z=x+yi\), where \(x\) and \(y\) are real numbers:
• abs() – absolute value
The absolute value or modulus of \(z\) is denoted \(|z|\) and is defined by:
$$ abs(z) = |z| = \sqrt{x^2 + y^2}.$$
• arg() – argument
For any non-zero \(|z|\) we can write the following expression:
$$ z = \left| z \right| e^{i \cdot arg (z)},$$
where \(arg (z)\) is the principal value of the argument of \(z\). It is chosen to be the unique value of the argument that lies within the interval \((–π, π]\). For \(z = 0\) by definition \(arg (0) = 0.\)
• inverse() – multiplicative inverse
The reciprocal (or multiplicative inverse) of every nonzero complex number \(z\) is a complex number \({ z }^{ -1 } \) such that \( z { z }^{ -1 } = 1\). So, we can easily get for any non-zero \(z\):
$${ z }^{ -1 }=\frac { 1 }{ z } =\frac { (x-yi) }{ (x+yi)(x-yi) } =\frac { (x-yi) }{ { x }^{ 2 }+{ y }^{ 2 } } =\frac { x }{ { x }^{ 2 }+{ y }^{ 2 } } -\frac { y }{ { x }^{ 2 }+{ y }^{ 2 } } i,$$
since non-zero \(z\) implies that \(x^{2}+y^{2}\) is greater than zero.
• sqrt() – square root
In complex analysis, any complex number \(z\) can be conventionally written as:
$$ z=|z|{ e }^{ i\varphi },$$
where \(-\pi \lt \varphi \le \pi \). Then we define the principal square root of \(z\) as follows:
$$ {\displaystyle {\sqrt {z}}={\sqrt {|z|}}e^{i\varphi /2}.}$$
The above formula can be expressed also in terms of trigonometric functions, using Euler’s formula:
$$ {\displaystyle {\sqrt {z}}={\sqrt {|z|}}\left(\cos {\frac {\varphi }{2}}+i\sin {\frac {\varphi }{2}}\right).}$$
• exp() – exponential function
Complex exponential function of \(z\) can be expressed in terms of trigonometric functions in the following way:
$${ e }^{ z }={ e }^{ x }{ e }^{ iy }={ e }^{ x }(cos(y)+i\cdot sin(y)).$$
• logₑ() – natural logarithmic function
For each nonzero complex number \(z\), the principal value of the logarithmic function \(logₑ(z)\) is the logarithm whose imaginary part lies in the interval \((−π, π]\). It can be expressed in the following way:
$$logₑ(z)=\ln |z|+i \cdot arg(z)=\ln {\sqrt {x^{2}+y^{2}}}+i \cdot {atan2} (y,x).$$
Related calculators
Check out our other math calculators such as Complex Numbers Calculator or Trigonometric Functions Calculator.