f ( t ) = L − 1 ( F ( s ) ) {f{{\left({t}\right)}}}={{L}}^{{-{1}}}{\left({F}{\left({s}\right)}\right)} f ( t ) = L − 1 ( F ( s ) ) F ( s ) = L ( f ( t ) ) {F}{\left({s}\right)}={L}{\left({f{{\left({t}\right)}}}\right)} F ( s ) = L ( f ( t ) ) 1 {1} 1 1 s \frac{{1}}{{s}} s 1 t n {{t}}^{{n}} t n n = 0 , 1 , 2 , 3 … {n}={0},{1},{2},{3}\ldots n = 0 , 1 , 2 , 3 … n ! s n + 1 \frac{{{n}!}}{{{{s}}^{{{n}+{1}}}}} s n + 1 n ! t n {{t}}^{{n}} t n n > − 1 {n}>-{1} n > − 1 Γ ( n + 1 ) s n + 1 \frac{{\Gamma{\left({n}+{1}\right)}}}{{{s}}^{{{n}+{1}}}} s n + 1 Γ ( n + 1 ) e a t {{e}}^{{{a}{t}}} e a t 1 s − a \frac{{1}}{{{s}-{a}}} s − a 1 t n − 1 2 {{t}}^{{{n}-\frac{{1}}{{2}}}} t n − 2 1 n = 1 , 2 , 3 … {n}={1},{2},{3}\ldots n = 1 , 2 , 3 … 1 ⋅ 3 ⋅ 5 ⋅ … ⋅ ( 2 n − 1 ) ⋅ π 2 n s n + 1 2 \frac{{{1}\cdot{3}\cdot{5}\cdot\ldots\cdot{\left({2}{n}-{1}\right)}\cdot\sqrt{{\pi}}}}{{{{2}}^{{n}}{{s}}^{{{n}+\frac{{1}}{{2}}}}}} 2 n s n + 2 1 1 ⋅ 3 ⋅ 5 ⋅ … ⋅ ( 2 n − 1 ) ⋅ π t \sqrt{{{t}}} t π 2 s 3 2 \frac{\sqrt{{\pi}}}{{{2}{{s}}^{{\frac{{3}}{{2}}}}}} 2 s 2 3 π sin ( a t ) {\sin{{\left({a}{t}\right)}}} sin ( a t ) a s 2 + a 2 \frac{{a}}{{{{s}}^{{2}}+{{a}}^{{2}}}} s 2 + a 2 a cos ( a t ) {\cos{{\left({a}{t}\right)}}} cos ( a t ) s s 2 + a 2 \frac{{s}}{{{{s}}^{{2}}+{{a}}^{{2}}}} s 2 + a 2 s sinh ( a t ) {\sinh{{\left({a}{t}\right)}}} sinh ( a t ) a s 2 − a 2 \frac{{a}}{{{{s}}^{{2}}-{{a}}^{{2}}}} s 2 − a 2 a cosh ( a t ) {\cosh{{\left({a}{t}\right)}}} cosh ( a t ) s s 2 − a 2 \frac{{s}}{{{{s}}^{{2}}-{{a}}^{{2}}}} s 2 − a 2 s t sin ( a t ) {t}{\sin{{\left({a}{t}\right)}}} t sin ( a t ) 2 a s ( s 2 + a 2 ) 2 \frac{{{2}{a}{s}}}{{{\left({{s}}^{{2}}+{{a}}^{{2}}\right)}}^{{2}}} ( s 2 + a 2 ) 2 2 a s t cos ( a t ) {t}{\cos{{\left({a}{t}\right)}}} t cos ( a t ) s 2 − a 2 ( s 2 + a 2 ) 2 \frac{{{{s}}^{{2}}-{{a}}^{{2}}}}{{{\left({{s}}^{{2}}+{{a}}^{{2}}\right)}}^{{2}}} ( s 2 + a 2 ) 2 s 2 − a 2 sin ( a t + b ) {\sin{{\left({a}{t}+{b}\right)}}} sin ( a t + b ) s ⋅ sin ( b ) + a ⋅ cos ( b ) s 2 + a 2 \frac{{{s}\cdot{\sin{{\left({b}\right)}}}+{a}\cdot{\cos{{\left({b}\right)}}}}}{{{{s}}^{{2}}+{{a}}^{{2}}}} s 2 + a 2 s ⋅ s i n ( b ) + a ⋅ c o s ( b ) cos ( a t + b ) {\cos{{\left({a}{t}+{b}\right)}}} cos ( a t + b ) s ⋅ cos ( b ) − a ⋅ sin ( b ) s 2 + a 2 \frac{{{s}\cdot{\cos{{\left({b}\right)}}}-{a}\cdot{\sin{{\left({b}\right)}}}}}{{{{s}}^{{2}}+{{a}}^{{2}}}} s 2 + a 2 s ⋅ c o s ( b ) − a ⋅ s i n ( b ) e a t sin ( b t ) {{e}}^{{{a}{t}}}{\sin{{\left({b}{t}\right)}}} e a t sin ( b t ) b ( s − a ) 2 + b 2 \frac{{b}}{{{{\left({s}-{a}\right)}}^{{2}}+{{b}}^{{2}}}} ( s − a ) 2 + b 2 b e a t cos ( b t ) {{e}}^{{{a}{t}}}{\cos{{\left({b}{t}\right)}}} e a t cos ( b t ) s − a ( s − a ) 2 + b 2 \frac{{{s}-{a}}}{{{{\left({s}-{a}\right)}}^{{2}}+{{b}}^{{2}}}} ( s − a ) 2 + b 2 s − a e a t sinh ( b t ) {{e}}^{{{a}{t}}}{\sinh{{\left({b}{t}\right)}}} e a t sinh ( b t ) b ( s − a ) 2 − b 2 \frac{{b}}{{{{\left({s}-{a}\right)}}^{{2}}-{{b}}^{{2}}}} ( s − a ) 2 − b 2 b e a t cosh ( b t ) {{e}}^{{{a}{t}}}{\cosh{{\left({b}{t}\right)}}} e a t cosh ( b t ) s − a ( s − a ) 2 − b 2 \frac{{{s}-{a}}}{{{{\left({s}-{a}\right)}}^{{2}}-{{b}}^{{2}}}} ( s − a ) 2 − b 2 s − a t n e a t {{t}}^{{n}}{{e}}^{{{a}{t}}} t n e a t n = 1 , 2 , 3 … {n}={1},{2},{3}\ldots n = 1 , 2 , 3 … n ! ( s − a ) n + 1 \frac{{{n}!}}{{{\left({s}-{a}\right)}}^{{{n}+{1}}}} ( s − a ) n + 1 n ! f ( c t ) {f{{\left({c}{t}\right)}}} f ( c t ) 1 c F ( s c ) \frac{{1}}{{c}}{F}{\left(\frac{{s}}{{c}}\right)} c 1 F ( c s ) u c ( t ) = u ( t − c ) {u}_{{c}}{\left({t}\right)}={u}{\left({t}-{c}\right)} u c ( t ) = u ( t − c ) e − c s s \frac{{{e}}^{{-{c}{s}}}}{{s}} s e − c s u c ( t ) f ( t − c ) {u}_{{c}}{\left({t}\right)}{f{{\left({t}-{c}\right)}}} u c ( t ) f ( t − c ) e − c s F ( s ) {{e}}^{{-{c}{s}}}{F}{\left({s}\right)} e − c s F ( s ) δ ( t − c ) \delta{\left({t}-{c}\right)} δ ( t − c ) e − c s {{e}}^{{-{c}{s}}} e − c s e c t f ( t ) {{e}}^{{{c}{t}}}{f{{\left({t}\right)}}} e c t f ( t ) F ( s − c ) {F}{\left({s}-{c}\right)} F ( s − c ) t n f ( t ) {{t}}^{{n}}{f{{\left({t}\right)}}} t n f ( t ) n = 1 , 2 , 3 … {n}={1},{2},{3}\ldots n = 1 , 2 , 3 … ( − 1 ) n F ( n ) ( s ) {{\left(-{1}\right)}}^{{n}}{{F}}^{{{\left({n}\right)}}}{\left({s}\right)} ( − 1 ) n F ( n ) ( s ) ∫ 0 t f ( τ ) d τ {\int_{{0}}^{{t}}}{f{{\left(\tau\right)}}}{d}\tau ∫ 0 t f ( τ ) d τ F ( s ) s \frac{{{F}{\left({s}\right)}}}{{s}} s F ( s ) ∫ 0 t f ( t − τ ) g ( τ ) d τ {\int_{{0}}^{{t}}}{f{{\left({t}-\tau\right)}}}{g{{\left(\tau\right)}}}{d}\tau ∫ 0 t f ( t − τ ) g ( τ ) d τ F ( s ) G ( s ) {F}{\left({s}\right)}{G}{\left({s}\right)} F ( s ) G ( s ) f ′ ( t ) {f{'}}{\left({t}\right)} f ′ ( t ) s F ( s ) − f ( 0 ) {s}{F}{\left({s}\right)}-{f{{\left({0}\right)}}} s F ( s ) − f ( 0 ) f ′ ′ ( t ) {f{''}}{\left({t}\right)} f ′′ ( t ) s 2 F ( s ) − s f ( 0 ) − f ′ ( 0 ) {{s}}^{{2}}{F}{\left({s}\right)}-{s}{f{{\left({0}\right)}}}-{f{'}}{\left({0}\right)} s 2 F ( s ) − s f ( 0 ) − f ′ ( 0 ) f ( n ) ( t ) {{f}}^{{{\left({n}\right)}}}{\left({t}\right)} f ( n ) ( t ) s n F ( s ) − ∑ k = 0 n − 1 ( s n − 1 − k f ( k ) ( 0 ) ) {{s}}^{{n}}{F}{\left({s}\right)}-{\sum_{{{k}={0}}}^{{{n}-{1}}}}{\left({{s}}^{{{n}-{1}-{k}}}{{f}}^{{{\left({k}\right)}}}{\left({0}\right)}\right)} s n F ( s ) − ∑ k = 0 n − 1 ( s n − 1 − k f ( k ) ( 0 ) ) 