Power Series Expansion for Exponential of Cosine of x/Proof 1

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Theorem

$e^{\cos x} = e \left({1 - \dfrac {x^2} 2 + \dfrac {x^4} 6 - \dfrac {31 x^6} {720} + \cdots}\right)$

for all $x \in \R$.


Proof

Let $f \left({x}\right) = e^{\cos x}$.

Then:


\(\displaystyle \frac \d {\d x} f \left({x}\right)\) \(=\) \(\displaystyle -\sin x \, e^{\cos x}\) Chain Rule
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d^2} {\d x^2} f \left({x}\right)\) \(=\) \(\displaystyle -\sin x \frac \d {\d x} e^{\cos x} + e^{\cos x} \frac \d {\d x} \left({-\sin x}\right)\) Product Rule
\(\displaystyle \) \(=\) \(\displaystyle \sin^2 x \, e^{\cos x} - \cos x \, e^{\cos x}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sin^2 x - \cos x}\right) e^{\cos x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d^3} {\d x^3} f \left({x}\right)\) \(=\) \(\displaystyle \left({\sin^2 x - \cos x}\right) \frac \d {\d x} e^{\cos x} + e^{\cos x} \frac \d {\d x} \left({\sin^2 x - \cos x}\right)\) Product Rule
\(\displaystyle \) \(=\) \(\displaystyle \left({\sin^2 x - \cos x}\right) \left({-\sin x}\right) e^{\cos x} + e^{\cos x} \left({2 \cos x \sin x + \sin x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({-\sin^3 x + 3 \cos x \sin x + \sin x}\right) e^{\cos x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d^4} {\d x^4} f \left({x}\right)\) \(=\) \(\displaystyle \left({-\sin^3 x + 3 x \cos x \sin + \sin x}\right) \frac \d {\d x} e^{\cos x} + e^{\cos x} \frac \d {\d x} \left({-\sin^3 x + 3 \cos x \sin x + \sin x}\right)\) Product Rule
\(\displaystyle \) \(=\) \(\displaystyle \left({-\sin^3 x + 3 \cos x \sin x + \sin x}\right) \left({-\sin x}\right) \, e^{\cos x} + e^{\cos x} \left({-3 \cos x \sin^2 x + 3 \left({\cos^2 x - \sin^2 x}\right) + \cos x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sin^4 x - 3 \cos x \sin^2 x - \sin^2 x - 3 \cos x \sin^2 x + 3 \left({\cos^2 x - \sin^2 x}\right) + \cos x}\right) e^{\cos x}\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x}\right) e^{\cos x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d^5} {\d x^5} f \left({x}\right)\) \(=\) \(\displaystyle \left({\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x}\right) \frac \d {\d x} e^{\cos x}\) Product Rule
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle e^{\cos x} \frac \d {\d x} \left({\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sin^4 x - 6 \cos x \sin^2 x + 3 \cos^2 x - 4 \sin^2 x + \cos x}\right) \left({-\sin x}\right) \, e^{\cos x}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle e^{\cos x} \left({4 \cos x \sin^3 x - 6 \left({2 \cos^2 x \sin x - \sin^3 x}\right) - 6 \cos x \sin x - 8 \cos x \sin x - \sin x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x}\right) e^{\cos x}\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac {\d^6} {\d x^6} f \left({x}\right)\) \(=\) \(\displaystyle \left({-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x}\right) \frac \d {\d x} e^{\cos x}\) Product Rule
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle e^{\cos x} \frac \d {\d x} \left({-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({-\sin^5 x + 10 \cos x \sin^3 x - 15 \cos^2 x \sin x + 6 \sin^3 x - 15 \cos x \sin x - \sin x}\right) \left({-\sin x}\right) \, e^{\cos x}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle e^{\cos x} \left({-5 \cos x \sin^4 x + 10 \left({3 \cos^2 x \sin^2 x - \sin^4 x}\right) - 15 \left({\cos^3 x - 2 \cos x \sin^2 x}\right) - 15 \left({\cos^2 x - \sin^2 x}\right) - \cos x}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({\sin^6 x - 15 \cos x \sin^4 x + 45 \cos^2 x \sin^2 x - 16 \sin^4 x - 15 \cos^3 x + 45 \cos x \sin^2 x - 15 \cos^2 x + 16 \sin^2 x - \cos x}\right) \, e^{\cos x}\)


By definition of Taylor series:

$f \left({x}\right) \sim \displaystyle \sum_{n \mathop = 0}^\infty \frac {\left({x - \xi}\right)^n} {n!} f^{\left({n}\right)} \left({\xi}\right)$


and so expanding about $\xi = 0$:

\(\displaystyle e^{\cos x}\) \(=\) \(\displaystyle \frac {x^0} {0!} e^{\cos 0} + \frac {x^1} {1!} \left({-\sin 0}\right) \, e^{\cos 0} + \frac {x^2} {2!} \left({\sin^2 0 - \cos 0}\right) e^{\cos 0}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac {x^3} {3!} \left({-\sin^3 0 + 3 \cos 0 \sin 0 + \sin 0}\right) e^{\cos 0} + \frac {x^4} {4!} \left({\sin^4 0 - 6 \cos 0 \sin^2 0 + 3 \cos^2 0 - 4 \sin^2 0 + \cos 0}\right) e^{\cos 0}\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac {x^5} {5!} \left({-\sin^5 0 + 10 \cos 0 \sin^3 0 - 15 \cos^2 0 \sin 0 + 6 \sin^3 0 - 15 \cos 0 \sin 0 - \sin 0}\right) e^{\cos 0} + \cdots\)
\(\displaystyle \) \(\) \(\, \displaystyle + \, \) \(\displaystyle \frac {x^6} {6!} \left({\sin^6 0 - 15 \cos 0 \sin^4 0 + 45 \cos^2 0 \sin^2 0 - 16 \sin^4 0 - 15 \cos^3 0 + 45 \cos 0 \sin^2 0 - 15 \cos^2 0 + 16 \sin^2 0 - \cos 0}\right) e^{\cos 0} + \cdots\)
\(\displaystyle \) \(=\) \(\displaystyle e \left({1 + 0 \times x - \frac {x^2} 2 + \frac {x^3} 6 \left({-0 + 0 + 0}\right) + \frac {x^4} {24} \left({0 - 0 + 3 - 0 + 1}\right) + \frac {x^5} {120} \left({-0 + 0 - 0 + 0 - 0 - 0}\right) + \frac {x^6} {720} \left({0 - 0 + 0 - 0 - 15 + 0 - 15 + 0 - 1}\right) + \cdots}\right)\) Sine of Zero is Zero, Exponential of Zero, Cosine of Zero is One
\(\displaystyle \) \(=\) \(\displaystyle e \left({1 + \frac {x^2} 2 - \frac {x^4} 6 - \frac {31 x^6} {720} + \cdots}\right)\)

$\blacksquare$