# ラプラス変換の線形性

$a$  , $b$ を実数の定数とすると

$\mathcal{L}\text{\hspace{0.17em}}\left\{af\left(t\right)\right\}=a\text{\hspace{0.17em}}\mathcal{L}\text{\hspace{0.17em}}\left\{f\left(t\right)\right\}$

$\mathcal{L}\text{}\left\{a{f}_{1}\left(t\right)+b{f}_{2}\left(t\right)\right\}$$=a\text{}\mathcal{L}\text{}\left\{{f}_{1}\left(t\right)\right\}+b\text{}\mathcal{L}\text{}\left\{{f}_{2}\left(t\right)\right\}$

の2式が成り立ち，ラプラス変換線形性を持つ．

## ■証明

$\mathcal{L}\text{\hspace{0.17em}}\left\{af\left(t\right)\right\}={\int }_{0}^{\infty }{e}^{-st}\left\{af\left(t\right)\right\}dt$

$=a{\int }_{0}^{\infty }{e}^{-st}f\left(t\right)dt$

$=a\text{\hspace{0.17em}}\mathcal{L}\text{\hspace{0.17em}}\left\{f\left(t\right)\right\}$

$\mathcal{L}\text{\hspace{0.17em}}\left\{a{f}_{1}\left(t\right)+b{f}_{2}\left(t\right)\right\}\text{\hspace{0.17em}}$$={\int }_{0}^{\infty }{e}^{-st}\left\{a{f}_{1}\left(t\right)+b{f}_{2}\left(t\right)\right\}dt$

$=a{\int }_{0}^{\infty }{e}^{-st}{f}_{1}\left(t\right)dt+b{\int }_{0}^{\infty }{e}^{-st}{f}_{2}\left(t\right)dt$

$=a\text{\hspace{0.17em}}\mathcal{L}\text{\hspace{0.17em}}\left\{{f}_{1}\left(t\right)\right\}+b\text{\hspace{0.17em}}\mathcal{L}\text{\hspace{0.17em}}\left\{{f}_{2}\left(t\right)\right\}$

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