Complex Functions

★★★★☆Undergraduate

📖Definition

A complex function maps complex numbers to complex numbers. f(z) = u(x,y) + iv(x,y) decomposes into real and imaginary parts.

f(z) = f(x + iy) = u(x,y) + iv(x,y)

Decomposition of complex function

e^z = e^x(cos y + isin y)

Complex exponential

sin z = \frace^iz - e^-iz2i, cos z = \frace^iz + e^-iz2

Complex trigonometric functions

예제 1

Find real and imaginary parts of f(z) = z².

예제 2

What is e^(iπ)?

Discovered by: Leonhard Euler (1748)

Euler discovered e^(iθ) = cos θ + i sin θ, laying foundations for complex analysis.

AC circuit analysis

Fourier transform

Wave functions

#복소함수#오일러#complex function#Euler