Complex Numbers Online Calculator

Professional online calculators for calculations with complex numbers z = x + yi

Basic Functions

Absolute Value (Modulus)
Calculates |z| - the length of the vector in the complex plane
Angle (Argument)
Calculates arg(z) - the angle to the positive real axis
Polar Form
Converts z = x + yi to polar form z = r·e^(iφ)
Conjugate
Reflection across the real axis: z̄ = x - yi
Multiplication
Multiplies two complex numbers z₁ · z₂
Division
Divides two complex numbers z₁ / z₂
Power
Calculates z^n for arbitrary exponents n
Exponential Function
Calculates e^z using Euler's formula
Square Root
Calculates √z - the principal root of the complex number
n-th Root
Calculates ⁿ√z - all n roots of a complex number
Reciprocal
Calculates 1/z - the reciprocal of the complex number
Natural Logarithm
Calculates ln(z) - natural logarithm to base e
Logarithm Base 10
Calculates log₁₀(z) - common logarithm

Trigonometric Functions

Sine (sin)
Calculates sin(z) for complex arguments
Cosine (cos)
Calculates cos(z) for complex arguments
Tangent (tan)
Calculates tan(z) = sin(z)/cos(z) with poles
Arc Sine (asin)
Inverse function of sine: arcsin(z)
Arc Cosine (acos)
Inverse function of cosine: arccos(z)
Arc Tangent (atan)
Inverse function of tangent: arctan(z)

Hyperbolic Functions

Hyperbolic Sine (sinh)
Calculates sinh(z) = (e^z - e^(-z))/2
Hyperbolic Cosine (cosh)
Calculates cosh(z) = (e^z + e^(-z))/2 - the catenary curve
Hyperbolic Tangent (tanh)
Calculates tanh(z) = sinh(z)/cosh(z) - sigmoid function

Special Functions

Airy Function Ai(z)
Solution of the Airy differential equation - important in quantum mechanics
Derivative Airy Function Ai'(z)
First derivative of the Airy function

Bessel Functions

Bessel-J (First Kind)
Jᵥ(z) - Bessel function of the first kind, order v
Bessel-Y (Second Kind)
Yᵥ(z) - Bessel function of the second kind (Neumann function)
Bessel-I (Modified, First Kind)
Iᵥ(z) - Modified Bessel function of the first kind
Bessel-K (Modified, Second Kind)
Kₙ(x) - Modified Bessel function of the second kind
Bessel-Je (Scaled)
Exponentially scaled Bessel-J function
Bessel-Ye (Scaled)
Exponentially scaled Bessel-Y function
Bessel-Ie (Scaled)
Exponentially scaled modified Bessel-I function
Bessel-Ke (Scaled)
Exponentially scaled modified Bessel-K function

About Complex Numbers

Complex numbers extend the real numbers by introducing the imaginary unit i with i² = -1. They have the form z = x + yi, where x is the real part and y is the imaginary part. These calculators help with understanding and calculating:

  • Representations - Cartesian, polar, exponential
  • Basic Operations - Addition, multiplication, division
  • Functions - Trigonometric, hyperbolic, exponential
  • Applications - Electrical engineering, quantum mechanics
  • Special Functions - Bessel, Airy, and more
  • Visualization - Complex plane (Argand diagram)
Important Formulas and Concepts
Euler's Formula
e^(ix) = cos(x) + i·sin(x)
Fundamental connection between exponential and trigonometric functions
Modulus and Argument
|z| = √(x² + y²), arg(z) = arctan(y/x)
Polar form: z = |z|·e^(i·arg(z))
Conjugate
z̄ = x - yi
Important: z·z̄ = |z|², Re(z) = (z+z̄)/2
De Moivre's Formula
(cos(θ) + i·sin(θ))ⁿ = cos(nθ) + i·sin(nθ)
For powers and roots of complex numbers
Tip: Complex numbers are fundamental in electrical engineering (AC circuit analysis), quantum mechanics (wave functions), signal processing (Fourier transform), and many other areas of physics and engineering.
Quick Reference
i² = -1
Imaginary Unit
z = 3+4i
Cartesian
|z| = 5
Modulus
z̄ = 3-4i
Conjugate
e^(iπ) = -1
Euler's Identity
Applications
Electrical Engineering:
AC circuit analysis with impedances
Quantum Mechanics:
Wave functions and operators
Signal Processing:
Fourier transform and filters
Fluid Dynamics:
Potential flows
History
16th Century:
Cardano uses √(-1) to solve cubic equations
1777:
Euler introduces the notation i
1806:
Gauss develops geometric interpretation
Today:
Essential in science and engineering
Related Calculator Categories
Mathematical Foundations
Applied Areas
All Categories