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L {∫t0f(t)dt}=F(s)s+f(−1)(0)s
ただし,f(−1)(0)=∫0−∞f(t)dt
L {1tf(t)}=∫∞sF(s)ds
一般に
L{∫t0∫t0∫t0⋯∫t0f(t)(dt)n}=F(s)sn+f(−1)(0)sn+f(−2)(0)sn−1+⋯+f(−n)(0)s
ただし
f(−n)(0)=∫0−∞∫0−∞∫0−∞⋯∫0−∞f(t)(dt)n
L{1tnf(t)}=∫∞s∫∞s∫∞s⋯∫∞sF(s)(ds)n
L {∫t0f(t)dt}=∫∞0e−st{∫t0f(t)dt}dt
=∫∞0(−1se−st)'{∫t0f(t)dt}dt
=[−1se−st∫t0f(t)dt]∞0−∫∞0(−1se−st)ddt{∫t0f(t)dt}dt
=[−1se−st∫t0f(t)dt]∞0+1s∫∞0e−stf(t)dt
=1s∫∞0e−stf(t)dt
=F(s)s
∫∞sF(s)ds=∫∞s∫∞0e−stf(t)dtds
=∫∞0{∫∞se−stf(t)ds}dt
=∫∞0f(t)∫∞se−stdsdt
=∫∞0f(t)[−1te−st]∞sdt
=∫∞0f(t){0−(−1te−st)}dt
=∫∞0e−st{f(t)t}dt
=L {1tf(t)}
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最終更新日: 2023年6月6日