Formula of Complex Analysis

Step Function (Heaviside Function)

\[ \theta(t)=\mathrm{i}\int_{-\infty}^{\infty} \, \frac{\mathrm{d}z}{2\pi} \frac{\mathrm{e}^{ -\mathrm{i}zt }}{z+i\varepsilon } \quad(\varepsilon \to 0 ) \]

Laplace Transform

Properties of the unilateral Laplace transform

Property	Time domain	\(s\) domain
Linearity	\(a f(t) + b g(t)\)	\({\displaystyle aF(s)+bG(s)\ }\)
Frequency-domain derivative	\({\displaystyle tf(t)\ }\)	\({\displaystyle -F'(s)\ }\)
Frequency-domain general derivatives	\(t^{n}f(t)\)	\((-1)^{n}F^{(n)}(s)\)
General Derivative	\({\displaystyle f^{(n)}(t)\ }\)	\({\displaystyle s^{n}F(s)-\sum _{k=1}^{n}s^{n-k}f^{(k-1)}(0^{+})\ }\)
Frequency-domain integration	\({\displaystyle {\frac {1}{t}}f(t)\ }\)	\({\displaystyle \int _{s}^{\infty }F(\sigma )\,d\sigma \ }\)
Time-domain integration	\({\displaystyle \int _{0}^{t}f(\tau )\,d\tau =(u*f)(t)}\)	\({\displaystyle {1 \over s}F(s)}\)
Frequency shifting	\({\displaystyle e^{at}f(t)\ }\)	\({\displaystyle F(s-a)\ }\)
Time shifting	\({\displaystyle f(t-a)u(t-a)\ }\)	\({\displaystyle e^{-as}F(s)\ }\)
Time scaling	\({\displaystyle f(at)}\)	\({\displaystyle {\frac {1}{a}}F\left({s \over a}\right)}\)
Multiplication	\({\displaystyle f(t)g(t)}\)	\({\displaystyle {\frac {1}{2\pi i}}\lim _{T\to \infty }\int _{c-iT}^{c+iT}F(\sigma )G(s-\sigma )\,d\sigma \ }\)
Convolution	\({\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau }\)	\({\displaystyle F(s)\cdot G(s)\ }\)
Complex conjugation	\({\displaystyle f^{*}(t)}\)	\({\displaystyle F^{*}(s^{*})}\)
Cross-correlation	\({\displaystyle (f\star g)(t)=\int _{0}^{\infty }f(\tau )^{*}\,g(t+\tau )\,d\tau }\)	\({\displaystyle F^{*}(-s^{*})\cdot G(s)}\)
Periodic function	\({\displaystyle f(t)}\)	\({\displaystyle {1 \over 1-e^{-Ts}}\int _{0}^{T}e^{-st}f(t)\,dt}\)
Periodic summation	\(\({\displaystyle \sum _{n=0}^{\infty }f(t-Tn)u(t-Tn)}\quad{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}f(t-Tn)u(t-Tn)}\)\)	\({\displaystyle {\frac {1}{1-e^{-Ts}}}F(s)}\quad{\displaystyle {\frac {1}{1+e^{-Ts}}}F(s)}\)