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Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/390
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378
SOMMATION
Les
corrections
sont expliquées en
page de discussion
1.
o
1
+
a
Cos
.
t
Cos
.
u
Cos
.
v
1
+
a
2
Cos
.2
t
Cos
.2
u
Cos
.2
v
1.2
+
a
3
Cos
.3
t
Cos
.3
u
Cos
.3
v
1.2.3
+
…
{\displaystyle 1+{\frac {a\operatorname {Cos} .t\operatorname {Cos} .u\operatorname {Cos} .v}{1}}+{\frac {a^{2}\operatorname {Cos} .2t\operatorname {Cos} .2u\operatorname {Cos} .2v}{1.2}}+{\frac {a^{3}\operatorname {Cos} .3t\operatorname {Cos} .3u\operatorname {Cos} .3v}{1.2.3}}+\ldots }
=
+
1
4
{
e
a
Cos
.
(
t
+
u
+
v
)
.
Cos
.
[
a
Sin
.
(
t
+
u
+
v
)
]
+
e
a
Cos
.
(
t
+
u
−
v
)
.
Cos
.
[
a
Sin
.
(
t
+
u
−
v
)
]
+
e
a
Cos
.
(
u
+
v
−
t
)
.
Cos
.
[
a
Sin
.
(
u
+
v
−
t
)
]
+
e
a
Cos
.
(
v
+
t
−
u
)
.
Cos
.
[
a
Sin
.
(
v
+
t
−
u
)
]
}
.
{\displaystyle =+{\frac {1}{4}}\left\{{\begin{aligned}&e^{a\operatorname {Cos} .(t+u+v)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t+u+v)\right]\\\\+&e^{a\operatorname {Cos} .(t+u-v)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(t+u-v)\right]\\\\+&e^{a\operatorname {Cos} .(u+v-t)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(u+v-t)\right]\\\\+&e^{a\operatorname {Cos} .(v+t-u)}.\operatorname {Cos} .\left[a\operatorname {Sin} .(v+t-u)\right]\end{aligned}}\right\}.}
2.
o
a
Sin
.
t
Sin
.
u
Sin
.
v
1
+
a
2
Sin
.2
t
Sin
.2
u
Sin
.2
v
1.2
+
a
3
Sin
.3
t
Sin
.3
u
Sin
.3
v
1.2.3
+
…
{\displaystyle {\frac {a\operatorname {Sin} .t\operatorname {Sin} .u\operatorname {Sin} .v}{1}}+{\frac {a^{2}\operatorname {Sin} .2t\operatorname {Sin} .2u\operatorname {Sin} .2v}{1.2}}+{\frac {a^{3}\operatorname {Sin} .3t\operatorname {Sin} .3u\operatorname {Sin} .3v}{1.2.3}}+\ldots }
=
−
1
4
{
e
a
Cos
.
(
t
+
u
+
v
)
.
Sin
.
[
a
Sin
.
(
t
+
u
+
v
)
]
−
e
a
Cos
.
(
t
+
u
−
v
)
.
Sin
.
[
a
Sin
.
(
t
+
u
−
v
)
]
−
e
a
Cos
.
(
u
+
v
−
t
)
.
Sin
.
[
a
Sin
.
(
u
+
v
−
t
)
]
−
e
a
Cos
.
(
v
+
t
−
u
)
.
Sin
.
[
a
Sin
.
(
v
+
t
−
u
)
]
}
.
{\displaystyle =-{\frac {1}{4}}\left\{{\begin{aligned}&e^{a\operatorname {Cos} .(t+u+v)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(t+u+v)\right]\\\\-&e^{a\operatorname {Cos} .(t+u-v)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(t+u-v)\right]\\\\-&e^{a\operatorname {Cos} .(u+v-t)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(u+v-t)\right]\\\\-&e^{a\operatorname {Cos} .(v+t-u)}.\operatorname {Sin} .\left[a\operatorname {Sin} .(v+t-u)\right]\end{aligned}}\right\}.}
3.
o
1
+
a
Cos
.
3
t
1
+
a
2
Cos
.
3
2
t
1.2
+
a
3
Cos
.
3
3
t
1.2.3
+
…
{\displaystyle 1+{\frac {a\operatorname {Cos} .^{3}t}{1}}+{\frac {a^{2}\operatorname {Cos} .^{3}2t}{1.2}}+{\frac {a^{3}\operatorname {Cos} .^{3}3t}{1.2.3}}+\ldots }
=
+
1
4
{
e
a
Cos
.3
t
.
Cos
.
(
a
Sin
.3
t
)
+
3
e
a
Cos
.
t
.
Cos
.
(
a
Sin
.
t
)
}
.
{\displaystyle =+{\frac {1}{4}}\left\{e^{a\operatorname {Cos} .3t}.\operatorname {Cos} .(a\operatorname {Sin} .3t)+3e^{a\operatorname {Cos} .t}.\operatorname {Cos} .(a\operatorname {Sin} .t)\right\}.}
4.
o
a
Sin
.
3
t
1
+
a
2
Sin
.
3
2
t
1.2
+
a
3
Sin
.
3
3
t
1.2.3
+
…
{\displaystyle {\frac {a\operatorname {Sin} .^{3}t}{1}}+{\frac {a^{2}\operatorname {Sin} .^{3}2t}{1.2}}+{\frac {a^{3}\operatorname {Sin} .^{3}3t}{1.2.3}}+\ldots }
=
−
1
4
{
e
a
Cos
.3
t
.
Sin
.
(
a
Sin
.3
t
)
−
3
e
a
Cos
.
t
.
Sin
.
(
a
Sin
.
t
)
}
.
{\displaystyle =-{\frac {1}{4}}\left\{e^{a\operatorname {Cos} .3t}.\operatorname {Sin} .(a\operatorname {Sin} .3t)-3e^{a\operatorname {Cos} .t}.\operatorname {Sin} .(a\operatorname {Sin} .t)\right\}.}
et ainsi de suite.