Sawtooth wave

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Codomain

( − 1 , 1 ) {\displaystyle \left(-1,1\right)}

Domain

R ∖ { n − 1 2 } , n ∈ Z {\displaystyle \mathbb {R} \setminus \left\{n-{\tfrac {1}{2}}\right\},n\in \mathbb {Z} }

Fields of application

Electronics, synthesizers

Fourier series

x ( t ) = − 2 π ∑ k = 1 ∞ ( − 1 ) k k sin ⁡ ( 2 π k t ) {\displaystyle x(t)=-{\frac {2}{\pi }}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{k}}{k}}\sin \left(2\pi kt\right)}

General definition

x ( t ) = 2 ( t − ⌊ t + 1 2 ⌋ ) , t − 1 2 ∉ Z {\displaystyle x(t)=2\left(t-\left\lfloor t+{\tfrac {1}{2}}\right\rfloor \right),t-{\tfrac {1}{2}}\notin \mathbb {Z} }

Parity

Odd