inverselaplace s/(s^2+a^2)

解

逆ラプラス

\frac{s}{s ^{2} + a ^{2}}

cos (t a^{2})

解答ステップ

L^{- 1} {\frac{s}{s ^{2} + a ^{2}}}

逆変換ルールを適用:

L^{- 1} {F (s)} = f (t)

ならば

L^{- 1} {s F (s)} = \frac{d}{d t} f (t) + f (0)

\frac{s}{s ^{2} + a ^{2}}

向け

: F (s) = \frac{1}{s ^{2} + a ^{2}}

= \frac{d}{d t} (L^{- 1} {\frac{1}{s ^{2} + a ^{2}}}) + L^{- 1} {\frac{1}{s ^{2} + a ^{2}}} (0)

L^{- 1} {\frac{1}{s ^{2} + a ^{2}}} : \frac{sin ( t a ^{2} )}{a ^{2}}

\frac{d}{d t} \frac{sin ( t a ^{2} )}{a ^{2}} = cos (t a^{2})

\frac{sin ( t a ^{2} )}{a ^{2}}

を

0 = 0

で評価する

= cos (t a^{2})