9.2: Useful Mathematical Identities
e jθ , cos θ , and sin θ
\[e^{jθ} = \lim_{n\to \infty} \Bigl(1 + j\frac {θ} {n}\Bigr)^n = \sum_{n=0}^{\infty} {\frac {1} {n!}} (jθ)^n = \cos θ +j\sin θ \nonumber \]
\[\cos θ = \sum_{n=0}^{\infty} \frac {(-1)^n} {(2n)!} θ^{2n} \nonumber \]
\[\sin θ = \sum_{n=0}^{\infty} \frac {(-1)^n} {(2n+1)!} θ^{2n+1} \nonumber \]
Trigonometric Identities
\[\sin^2 {θ} + \cos^2 {θ} = 1 \nonumber \]
\[\sin (θ+φ) = \sin {θ} \cos {φ} + \cos {θ} \sin {φ} \nonumber \]
\[\cos (θ+φ) = \cos {θ} \cos {φ} - \sin {θ} \sin {φ} \nonumber \]
\[\sin (θ-φ) = \sin {θ} \cos {φ} - \cos {θ} \sin {φ} \nonumber \]
\[\cos (θ-φ) = \cos {θ} \cos {φ} + \sin {θ} \sin {φ} \nonumber \]
Euler's Equations
\[e^{jθ} = \cos θ + j\sin θ \nonumber \]
\[\sin θ = \frac {e^{jθ}-e^{-jθ}} {2j} \nonumber \]
\[\cos θ = \frac {e^{jθ}+e^{-jθ}} {2} \nonumber \]
De Moivre's Identity
\[(\cos θ +j\sin θ)^n = \cos {nθ} + j\sin {nθ} \nonumber \]
Binomial Expansion
\[(x+y)^N = \sum_{n=0}^{N} {\Bigl(\frac {N!} {(N-n)!n!} \Bigr)x^n y^{N-n}} \nonumber \]
\[2^N = \sum_{n=0}^{N} {\Bigl(\frac {N!} {(N-n)!n!} \Bigr)} \nonumber \]
Geometric Sums
\[\sum_{k=0}^{\infty} {az^k} = \frac {a} {1-z} ;|z| \lt 1 \nonumber \]
\[\sum_{k=0}^{N-1} {az^k} = \frac {a\left(1-z^N\right)} {1-z} ;z \neq 1 \nonumber \]
Taylor's Series
\[f(x) = \sum_{k=0}^{\infty} {f^{(k)}(a)\frac {(x-a)^k} {k!}} \nonumber \]
(Maclaurin's Series if a=0)