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Mathematics
Mathematics
Problem
Let
g
(
x
)
=
e
a
x
sin
(
b
x
)
g(x)=e^{ax}\sin(bx)
g
(
x
)
=
e
a
x
sin
(
b
x
)
, where
a
a
a
and
b
b
b
are real constants, not both zero. Define
R
=
a
2
+
b
2
R=\sqrt{a^{2}+b^{2}}
R
=
a
2
+
b
2
and choose an angle
θ
\theta
θ
satisfying
cos
θ
=
a
R
\cos\theta=\dfrac{a}{R}
cos
θ
=
R
a
and
sin
θ
=
b
R
\sin\theta=\dfrac{b}{R}
sin
θ
=
R
b
. For a positive integer
n
n
n
, derive a closed-form expression for
g
(
n
)
(
x
)
g^{(n)}(x)
g
(
n
)
(
x
)
in terms of
R
R
R
,
θ
\theta
θ
,
n
n
n
, and
x
x
x
.
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