Euler's formula 歐拉公式

Leonhard Euler (1707-1783) 瑞士數學家
Euler 歐拉/尤拉

Euler's formula helps to find the exponential function of a complex number.

Euler's formula 歐拉公式:

e^jx = cosx + isinx = cisx

Therefore,

e^iθ = cosθ + isinθ (1)
e^-iθ = cos(-θ) + isin(-θ) = cosθ - isinθ (2)

(1) + (2) gives
=> e^iθ + e^-iθ =  2cosθ
=> cosθ = ( e^iθ + e^-iθ )/2

(1) - (2) gives
=> e^jθ - e^-jθ = 2isinθ
=> sinθ = ( e^iθ - e^-iθ )/2i

cosθ = Re(e^iθ) =( e^iθ + e^-iθ )/2
sinθ = Im(e^iθ) =( e^iθ - e^-iθ )/2i

Example:

eⁱ = cos(1) + isin(1) = 0.540302306 + 0.841470985i
e^iπ = cos(π) + isin(π) = -1 => e^iπ + 1 = 0

Matlab function:
exp(i*pi) or exp(j*pi)

參考資料

cis (mathematics) (Wikipedia)
Relations between cosine, sine and exponential functions
Exponential function & natural logarithm 指數函數與自然對數
Magnitude of Exponential Function exp(iθ) = ? 如何求exp(iθ)的大小/絕對值