# 制御と振動の数学/第一類/Laplace 変換/Laplace 変換の定義とその基本的性質/合成積の Laplace 変換

(2.3)
${\displaystyle \int _{0}^{t}f(t-\tau )g(\tau )d\tau }$

${\displaystyle f(t)}$${\displaystyle g(t)}$合成積といい，

${\displaystyle f(t)*g(t)}$ または ${\displaystyle f*g}$

と略記する[1]． 次の性質は重要である．

(2.4)
${\displaystyle {\mathcal {L}}[f*g]={\mathcal {L}}[f]\cdot {\mathcal {L}}[g]}$

${\displaystyle {\mathcal {L}}[f*g]=\int _{0}^{\infty }\left\{\int _{0}^{t}f(t-\tau )g(\tau )d\tau \right\}e^{-st}dt}$

${\displaystyle {\mathcal {L}}[f*g]=\int _{0}^{\infty }\left\{\int _{\tau }^{\infty }f(t-\tau )e^{-st}dt\right\}g(\tau )d\tau }$

となる．ここで ${\displaystyle e^{-st}=e^{-s(t-\tau )}\cdot e^{-s\tau }}$ と変形し，${\displaystyle v:=t-\tau }$ によって、積分変数を ${\displaystyle t}$ から ${\displaystyle v}$ に変えると，

${\displaystyle {\mathcal {L}}[f*g]=\int _{0}^{\infty }\left\{\int _{0}^{\infty }f(v)e^{-sv}dv\right\}g(\tau )e^{-s\tau }d\tau }$[2]
${\displaystyle {\mathcal {L}}[f*g]=\int _{0}^{\infty }f(v)e^{-sv}dv\int _{0}^{\infty }g(\tau )e^{-s\tau }d\tau }$
${\displaystyle ={\mathcal {L}}[f]\cdot {\mathcal {L}}[g]}$

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

と積分の上限を ${\displaystyle \infty }$ にとることができる[3]． このようにしておいてから積分順序を交換すると，

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

となる．ここで再び ${\displaystyle f(t)=0\quad (t<0)}$ を想起すると，内側の積分の下限は ${\displaystyle \tau }$ でよく[4]

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

を得る．[5]

(2.4a)
${\displaystyle \int _{0}^{T}dt\int _{t}^{T}d\tau f(t-\tau )g(\tau )e^{-st}=\int _{0}^{T}d\tau \int _{0}^{\tau }dt\ f(t-\tau )g(\tau )e^{-st}}$

(2.4a)の左辺と ${\displaystyle S_{1}}$ を加えたものは，

${\displaystyle S_{1}+\int _{0}^{T}dt\int _{t}^{T}d\tau f(t-\tau )g(\tau )e^{-st}=\int _{0}^{T}dt\left\{\int _{0}^{t}d\tau f(t-\tau )g(\tau )e^{-st}+\int _{t}^{T}d\tau f(t-\tau )g(\tau )e^{-st}\right\}}$
${\displaystyle =\int _{0}^{T}dt\int _{0}^{T}d\tau f(t-\tau )g(\tau )e^{-st}}$

また，式(2.4a)の右辺と ${\displaystyle S_{2}}$ を加えたものは，

${\displaystyle S_{2}+\int _{0}^{T}d\tau \int _{0}^{\tau }dt\ f(t-\tau )g(\tau )e^{-st}=\int _{0}^{T}d\tau \left\{\int _{\tau }^{T}dt\ f(t-\tau )g(\tau )e^{-st}+\int _{0}^{\tau }dt\ f(t-\tau )g(\tau )e^{-st}\right\}}$
${\displaystyle =\int _{0}^{T}d\tau \int _{0}^{T}dt\ f(t-\tau )g(\tau )e^{-st}}$

${\displaystyle S_{1}+\int _{0}^{T}dt\int _{t}^{T}d\tau f(t-\tau )g(\tau )e^{-st}=S_{2}+\int _{0}^{T}d\tau \int _{0}^{\tau }dt\ f(t-\tau )g(\tau )e^{-st}}$

よって，式(2.4a)より，${\displaystyle S_{1}=S_{2}}$，すなわち，

${\displaystyle \int _{0}^{T}dt\int _{0}^{t}d\tau f(t-\tau )g(\tau )e^{-st}=\int _{0}^{T}d\tau \int _{\tau }^{T}dt\ f(t-\tau )g(\tau )e^{-st}}$

${\displaystyle T\to \infty }$ で両辺とも極限値を持てば，同じくこの等式は成立する．

${\displaystyle \diamondsuit }$

(i) ${\displaystyle f*g=g*f}$

(ii) ${\displaystyle (kf)*g=k(f*g)}$

(iii) ${\displaystyle f*(g*h)=(f*g)*h}$

(iv) ${\displaystyle f*(g+h)=f*g+f*h}$

を示せ．

(i)

${\displaystyle f*g=\int _{0}^{t}f(t-\tau )g(\tau )d\tau }$

にて，${\displaystyle v=t-\tau }$ とおいて積分変数を ${\displaystyle \tau }$ から ${\displaystyle v}$ に換えるとき，${\displaystyle dv=-d\tau }$，また ${\displaystyle \tau }$${\displaystyle 0\to t}$ と変化するとき ${\displaystyle v}$${\displaystyle t\to 0}$ と変化するから，

${\displaystyle f*g=-\int _{t}^{0}f(v)g(t-v)dv=\int _{0}^{t}g(t-v)f(v)dv=g*f}$

(ii)

${\displaystyle (kf)*g=\int _{0}^{t}\{kf(t-\tau )\}g(\tau )d\tau }$
${\displaystyle =\int _{0}^{t}kf(t-\tau )g(\tau )d\tau }$
${\displaystyle =k\int _{0}^{t}f(t-\tau )g(\tau )d\tau =k(f*g)}$

(iii) これはとても難しい…いつか分かる日が来るのだろうか？

(iv)

${\displaystyle f*(g+h)=\int _{0}^{t}f(t-\tau )\left\{g(\tau )+h(\tau )\right\}d\tau }$
${\displaystyle =\int _{0}^{t}f(t-\tau )g(\tau )d\tau +\int _{0}^{t}f(t-\tau )h(\tau )d\tau =f*g+f*h}$

${\displaystyle \diamondsuit }$

1. ^ この積分形に近い型としては，いわゆる「複数桁×複数桁の筆算」．積の一つの桁に着目すると ${\displaystyle f(t-\tau )g(\tau )}$ 形の総和をとる．
2. ^ ${\displaystyle v=t-\tau }$ の両辺を ${\displaystyle t}$ で微分すると ${\displaystyle {\frac {dv}{dt}}=1\therefore {\frac {dt}{dv}}=1}$．よって ${\displaystyle dt={\frac {dt}{dv}}dv}$ すなわち ${\displaystyle dt=dv}$にて積分変数を ${\displaystyle t}$ から ${\displaystyle v}$ に変更できる．また積分範囲は ${\displaystyle t}$${\displaystyle \tau }$ から ${\displaystyle \infty }$ に動くとき ${\displaystyle v}$${\displaystyle 0}$ から ${\displaystyle \infty }$ に動く．
3. ^ なぜならば，内側の積分変数 ${\displaystyle \tau }$ による積分で，${\displaystyle \tau >t}$ ならば ${\displaystyle t-\tau <0}$ よって ${\displaystyle f(t-\tau )=0}$
${\displaystyle \therefore \int _{t}^{\infty }f(t-\tau )g(\tau )d\tau =\int _{t}^{\infty }0\cdot g(\tau )d\tau =0}$
${\displaystyle \therefore \int _{0}^{t}d\tau =\int _{0}^{t}d\tau +\int _{t}^{\infty }d\tau =\int _{0}^{\infty }d\tau }$
4. ^ 積分変数${\displaystyle t}$${\displaystyle 0 の範囲のとき ${\displaystyle t-\tau <0\therefore f(t-\tau )=0}$
5. ^ この続きは上記のとおり ${\displaystyle e^{-st}=e^{-s(t-\tau )}\cdot e^{-s\tau }}$ と変形し，${\displaystyle v:=t-\tau \therefore dv=dt}$ として積分変数を ${\displaystyle t}$ から ${\displaystyle v}$ に変える．