Power Series Expansion for Exponential of x by Cosine of x
Theorem
\(\ds e^x \cos x\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {2^{n / 2} \, \map \cos {n \pi / 4} x^n} {n!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + x - \frac {x^3} 3 - \frac {x^4} 6 + \cdots\) |
for all $x \in \R$.
Proof
Let $\map f x = e^x \cos x$.
By definition of Maclaurin series:
- $(1): \quad \map f x \sim \ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!} \map {f^{\paren n} } 0$
where $\map {f^{\paren n} } 0$ denotes the $n$th derivative of $f$ with respect to $x$ evaluated at $x = 0$.
It remains to be shown that:
- $\map {f^{\paren n} } 0 = 2^{n / 2} \map \cos {\dfrac {n \pi} 4}$
The proof proceeds by induction.
For all $n \in \Z_{\ge 0}$, let $\map P n$ be the proposition:
- $\map {f^{\paren n} } x = 2^{n / 2} e^x \map \cos {x + \dfrac {n \pi} 4}$
Basis for the Induction
$\map P 0$ is the case:
\(\ds \map f x\) | \(=\) | \(\ds e^x \cos x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2^{0 / 2} e^x \map \cos {x + \dfrac {0 \pi} 4}\) |
Thus $\map P 0$ is seen to hold.
This is the basis for the induction.
Induction Hypothesis
Now it needs to be shown that, if $\map P k$ is true, where $k \ge 0$, then it logically follows that $\map P {k + 1}$ is true.
So this is the induction hypothesis:
- $\map {f^{\paren k} } x = 2^{k / 2} e^x \cos \left({x + \dfrac {k \pi} 4}\right)$
from which it is to be shown that:
- $\map {f^{\paren {k + 1} } } x = 2^{\paren {k + 1} / 2} e^x \map \cos {x + \dfrac {\paren {k + 1} \pi} 4}$
Induction Step
This is the induction step:
\(\ds \map {f^{\paren {k + 1} } } x\) | \(=\) | \(\ds \frac \d {\d x} \map {f^{\paren k} } x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac \d {\d x} 2^{k / 2} e^x \map \cos {x + \dfrac {k \pi} 4}\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^{k / 2} \paren {\map \cos {x + \dfrac {k \pi} 4} \frac \d {\d x} e^x + e^x \frac \d {\d x} \map \cos {x + \dfrac {k \pi} 4} }\) | Product Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^{k / 2} \paren {\map \cos {x + \dfrac {k \pi} 4} \frac \d {\d x} e^x - e^x \map \sin {x + \dfrac {k \pi} 4} }\) | Derivative of Cosine Function, Derivatives of Function of $a x + b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^{k / 2} e^x \paren {\map \cos {x + \dfrac {k \pi} 4} - \map \sin {x + \dfrac {k \pi} 4} }\) | Derivative of Exponential Function and simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^{k / 2} e^x \sqrt 2 \map \cos {x + \dfrac {k \pi} 4 + \dfrac \pi 4}\) | Cosine of x minus Sine of x: Cosine Form | |||||||||||
\(\ds \) | \(=\) | \(\ds 2^{\paren {k + 1} / 2} e^x \map \cos {x + \dfrac {\paren {k + 1} \pi} 4}\) | simplification |
So $\map P k \implies \map P {k + 1}$ and by the Principle of Mathematical Induction:
- $\forall n \in \Z_{\ge 0}: \map {f^{\paren n} } x = 2^{n / 2} e^x \map \cos {x + \dfrac {n \pi} 4}$
The result follows by setting $x = 0$ and substituting for $\map {f^{\paren n} } 0$ in $(1)$.
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 20$: Miscellaneous Series: $20.47$