#set document(title: "10.1 Table of Laplace Transforms", author: "Jiří Lebl")
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== 10.1#h(0.6em)Table of Laplace Transforms


    The function #math.equation(block: false, alt: "u")[$u$] is the Heaviside function, #math.equation(block: false, alt: "δ")[$δ$] is the Dirac delta function, and


    #math.equation(block: true, alt: "Γ open parenthesis t close parenthesis equals ∫ 0 infinity e to the power minus τ τ to the power t minus 1 d τ , erf open parenthesis t close parenthesis equals the fraction 2 over the square root of π ∫ 0 t e to the power minus τ squared d τ , erfc open parenthesis t close parenthesis equals 1 minus erf open parenthesis t close parenthesis .")[$Γ ( t ) = ∫_(0)^(∞) e^(− τ) τ^(t − 1) d τ , #h(2em) "erf" ( t ) = frac(2, sqrt(π)) ∫_(0)^(t) e^(− τ^(2)) d τ , #h(2em) "erfc" ( t ) = 1 − "erf" ( t ) .$]


    Table #math.equation(block: false, alt: "1")[$1$]


    #figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([#strong[#math.equation(block: false, alt: "f open parenthesis t close parenthesis")[$f ( t )$]]], [#strong[#math.equation(block: false, alt: "F open parenthesis s close parenthesis equals ℒ { f open parenthesis t close parenthesis } equals ∫ 0 infinity e to the power minus s t f open parenthesis t close parenthesis d t")[$F ( s ) = ℒ \{ f ( t ) \} = ∫_(0)^(∞) e^(− s t) f ( t ) d t$]]]),
  [#math.equation(block: false, alt: "C")[$C$]], [#math.equation(block: false, alt: "the fraction C over s")[$frac(C, s)$]],
  [#math.equation(block: false, alt: "t")[$t$]], [#math.equation(block: false, alt: "the fraction 1 over s squared")[$frac(1, s^(2))$]],
  [#math.equation(block: false, alt: "t squared")[$t^(2)$]], [#math.equation(block: false, alt: "the fraction 2 over s cubed")[$frac(2, s^(3))$]],
  [#math.equation(block: false, alt: "t to the power n")[$t^(n)$]], [#math.equation(block: false, alt: "the fraction n ! over s to the power n plus 1")[$frac(n !, s^(n + 1))$]],
  [#math.equation(block: false, alt: "t to the power p open parenthesis p greater than 0 close parenthesis")[$t^(p) #h(1em) ( p > 0 )$]], [#math.equation(block: false, alt: "the fraction Γ open parenthesis p plus 1 close parenthesis over s to the power p plus 1")[$frac(Γ ( p + 1 ), s^(p + 1))$]],
  [#math.equation(block: false, alt: "e to the power minus a t")[$e^(− a t)$]], [#math.equation(block: false, alt: "the fraction 1 over s plus a")[$frac(1, s + a)$]],
  [#math.equation(block: false, alt: "sin open parenthesis ω t close parenthesis")[$sin ( ω t )$]], [#math.equation(block: false, alt: "the fraction ω over s squared plus ω squared")[$frac(ω, s^(2) + ω^(2))$]],
  [#math.equation(block: false, alt: "cos open parenthesis ω t close parenthesis")[$cos ( ω t )$]], [#math.equation(block: false, alt: "the fraction s over s squared plus ω squared")[$frac(s, s^(2) + ω^(2))$]],
  [#math.equation(block: false, alt: "sinh open parenthesis ω t close parenthesis")[$sinh ( ω t )$]], [#math.equation(block: false, alt: "the fraction ω over s squared minus ω squared")[$frac(ω, s^(2) − ω^(2))$]],
  [#math.equation(block: false, alt: "cosh open parenthesis ω t close parenthesis")[$cosh ( ω t )$]], [#math.equation(block: false, alt: "the fraction s over s squared minus ω squared")[$frac(s, s^(2) − ω^(2))$]],
  [#math.equation(block: false, alt: "u open parenthesis t minus a close parenthesis")[$u ( t − a )$]], [#math.equation(block: false, alt: "the fraction e to the power minus a s over s")[$frac(e^(− a s), s)$]],
  [#math.equation(block: false, alt: "δ open parenthesis t close parenthesis")[$δ ( t )$]], [#math.equation(block: false, alt: "1")[$1$]],
  [#math.equation(block: false, alt: "δ open parenthesis t minus a close parenthesis")[$δ ( t − a )$]], [#math.equation(block: false, alt: "e to the power minus a s")[$e^(− a s)$]],
  [#math.equation(block: false, alt: "erf open parenthesis the fraction t over 2 a close parenthesis")[$"erf" ( frac(t, 2 a) )$]], [#math.equation(block: false, alt: "the fraction 1 over s e to the power open parenthesis a s close parenthesis squared erfc open parenthesis a s close parenthesis")[$frac(1, s) e^(( a s )^(2)) "erfc" ( a s )$]],
  [#math.equation(block: false, alt: "the fraction 1 over the square root of π t exp open parenthesis the fraction minus a squared over 4 t close parenthesis open parenthesis a greater than or equal to 0 close parenthesis")[$frac(1, sqrt(π t)) "exp" ( frac(− a^(2), 4 t) ) #h(1em) ( a ≥ 0 )$]], [#math.equation(block: false, alt: "the fraction e to the power minus a s over the square root of s")[$frac(e^(− a s), sqrt(s))$]],
  [#math.equation(block: false, alt: "the fraction 1 over the square root of π t minus a e to the power a squared t erfc open parenthesis a the square root of t close parenthesis open parenthesis a greater than 0 close parenthesis")[$frac(1, sqrt(π t)) − a e^(a^(2) t) "erfc" ( a sqrt(t) ) #h(1em) ( a > 0 )$]], [#math.equation(block: false, alt: "the fraction 1 over the square root of s plus a")[$frac(1, sqrt(s) + a)$]],
  [#math.equation(block: false, alt: "a f open parenthesis t close parenthesis plus b g open parenthesis t close parenthesis")[$a f ( t ) + b g ( t )$]], [#math.equation(block: false, alt: "a F open parenthesis s close parenthesis plus b G open parenthesis s close parenthesis")[$a F ( s ) + b G ( s )$]],
  [#math.equation(block: false, alt: "f open parenthesis a t close parenthesis open parenthesis a greater than 0 close parenthesis")[$f ( a t ) #h(1em) ( a > 0 )$]], [#math.equation(block: false, alt: "the fraction 1 over a F open parenthesis the fraction s over a close parenthesis")[$frac(1, a) F ( frac(s, a) )$]],
  [#math.equation(block: false, alt: "f open parenthesis t minus a close parenthesis u open parenthesis t minus a close parenthesis")[$f ( t − a ) u ( t − a )$]], [#math.equation(block: false, alt: "e to the power minus a s F open parenthesis s close parenthesis")[$e^(− a s) F ( s )$]],
  [#math.equation(block: false, alt: "e to the power minus a t f open parenthesis t close parenthesis")[$e^(− a t) f ( t )$]], [#math.equation(block: false, alt: "F open parenthesis s plus a close parenthesis")[$F ( s + a )$]],
  [#math.equation(block: false, alt: "g prime open parenthesis t close parenthesis")[$g^(′) ( t )$]], [#math.equation(block: false, alt: "s G open parenthesis s close parenthesis minus g open parenthesis 0 close parenthesis")[$s G ( s ) − g ( 0 )$]],
  [#math.equation(block: false, alt: "g double prime open parenthesis t close parenthesis")[$g^(″) ( t )$]], [#math.equation(block: false, alt: "s squared G open parenthesis s close parenthesis minus s g open parenthesis 0 close parenthesis minus g prime open parenthesis 0 close parenthesis")[$s^(2) G ( s ) − s g ( 0 ) − g^(′) ( 0 )$]],
  [#math.equation(block: false, alt: "g to the power ‴ open parenthesis t close parenthesis")[$g^(‴) ( t )$]], [#math.equation(block: false, alt: "s cubed G open parenthesis s close parenthesis minus s squared g open parenthesis 0 close parenthesis minus s g prime open parenthesis 0 close parenthesis minus g double prime open parenthesis 0 close parenthesis")[$s^(3) G ( s ) − s^(2) g ( 0 ) − s g^(′) ( 0 ) − g^(″) ( 0 )$]],
  [#math.equation(block: false, alt: "g to the power open parenthesis n close parenthesis open parenthesis t close parenthesis")[$g^(( n )) ( t )$]], [#math.equation(block: false, alt: "s to the power n G open parenthesis s close parenthesis minus s to the power n minus 1 g open parenthesis 0 close parenthesis minus ⋯ minus g to the power open parenthesis n minus 1 close parenthesis open parenthesis 0 close parenthesis")[$s^(n) G ( s ) − s^(n − 1) g ( 0 ) − ⋯ − g^(( n − 1 )) ( 0 )$]],
  [#math.equation(block: false, alt: "open parenthesis f * g close parenthesis open parenthesis t close parenthesis equals ∫ 0 t f open parenthesis τ close parenthesis g open parenthesis t minus τ close parenthesis d τ")[$( f * g ) ( t ) = ∫_(0)^(t) f ( τ ) g ( t − τ ) d τ$]], [#math.equation(block: false, alt: "F open parenthesis s close parenthesis G open parenthesis s close parenthesis")[$F ( s ) G ( s )$]],
  [#math.equation(block: false, alt: "t f open parenthesis t close parenthesis")[$t f ( t )$]], [#math.equation(block: false, alt: "minus F prime open parenthesis s close parenthesis")[$− F^(′) ( s )$]],
  [#math.equation(block: false, alt: "t to the power n f open parenthesis t close parenthesis")[$t^(n) f ( t )$]], [#math.equation(block: false, alt: "open parenthesis minus 1 close parenthesis to the power n F to the power open parenthesis n close parenthesis open parenthesis s close parenthesis")[$( − 1 )^(n) F^(( n )) ( s )$]],
  [#math.equation(block: false, alt: "∫ 0 t f open parenthesis τ close parenthesis d τ")[$∫_(0)^(t) f ( τ ) d τ$]], [#math.equation(block: false, alt: "the fraction 1 over s F open parenthesis s close parenthesis")[$frac(1, s) F ( s )$]],
  [#math.equation(block: false, alt: "the fraction f open parenthesis t close parenthesis over t")[$frac(f ( t ), t)$]], [#math.equation(block: false, alt: "∫ s infinity F open parenthesis σ close parenthesis d σ")[$∫_(s)^(∞) F ( σ ) d σ$]],
))