AcademyComplex Equations
Academy
Complex Functions
Level 1 - Math I (Physics) topic page in Complex Equations.
Principle
A function of a complex variable takes a complex input and returns a complex output. The main functions needed here are the exponential, trigonometric, and hyperbolic functions.
Complex input
\[z=x+iy,\quad x,y\in\mathbb R\]
The complex exponential is the central example because it connects growth, rotation, and periodicity.
Notation
- \(z=x+iy\) is a complex input.
- \(x=\operatorname{Re}(z)\) is the real part.
- \(y=\operatorname{Im}(z)\) is the imaginary part.
- \(e^z\) is the complex exponential.
- \(\sin z\), \(\cos z\), \(\sinh z\), and \(\cosh z\) are complex-valued functions.
Method
Write \(z=x+iy\) and split the exponential:
Split exponential
\[e^z=e^{x+iy}=e^xe^{iy}\]
Use Euler's formula for the imaginary exponential:
Euler substitution
\[e^{iy}=\cos y+i\sin y\]
Therefore:
Complex exponential form
\[e^z=e^x(\cos y+i\sin y)\]
This immediately gives:
Real part
\[\operatorname{Re}(e^z)=e^x\cos y\]
Imaginary part
\[\operatorname{Im}(e^z)=e^x\sin y\]
Modulus
\[|e^z|=e^x=e^{\operatorname{Re}(z)}\]
Argument
\[\arg(e^z)=y\quad\text{mod }2\pi\]
Rules
Define the trigonometric and hyperbolic functions using exponentials:
Cosine definition
\[\cos z=\frac{e^{iz}+e^{-iz}}{2}\]
Sine definition
\[\sin z=\frac{e^{iz}-e^{-iz}}{2i}\]
Hyperbolic cosine definition
\[\cosh z=\frac{e^z+e^{-z}}{2}\]
Hyperbolic sine definition
\[\sinh z=\frac{e^z-e^{-z}}{2}\]
Useful identities follow by substituting \(iz\):
Complex trig hyperbolic links
\[\cos(iz)=\cosh z,\quad \sin(iz)=i\sinh z\]
Complex hyperbolic trig links
\[\cosh(iz)=\cos z,\quad \sinh(iz)=i\sin z\]
Examples
Find the modulus of:
Example target
\[e^{(1+i)(2-i)}\]
First multiply the exponent:
Exponent product
\[(1+i)(2-i)=2-i+2i-i^2=3+i\]
So:
Example exponential
\[e^{(1+i)(2-i)}=e^{3+i}\]
The modulus of \(e^{x+iy}\) is \(e^x\), so:
Example modulus
\[|e^{3+i}|=e^3\]
Checks
- The exponential \(e^z\) is periodic in the imaginary direction: \(e^{z+2\pi i}=e^z\).
- The exponential never equals zero because \(|e^z|=e^{\operatorname{Re}(z)}\gt0\).
- The argument of \(e^z\) is only determined modulo \(2\pi\).
- Complex trigonometric and hyperbolic functions usually have complex values, even when their names are familiar from real calculus.