# Fourier Series/Exponential of x over Minus Pi to Pi

## Theorem

Let $\map f x$ be the real function defined on $\R$ as:

$\map f x$ and its $7$th approximation
$\map f x = \begin{cases} e^x & : -\pi < x \le \pi \\ \map f {x + 2 \pi} & : \text{everywhere} \end{cases}$

Then its Fourier series can be expressed as:

 $\ds \map f x$ $\sim$ $\ds \frac {\sinh \pi} \pi \paren {1 + 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {1 + n^2} \paren {\cos n x - n \sin n x} }$ $\ds$ $=$ $\ds \frac {\sinh \pi} \pi \paren {1 + 2 \paren {-\dfrac {\cos x - \sin x} 2 + \dfrac {\cos 2 x - 2 \sin 2 x} 5 - \dfrac {\cos 3 x - 3 \sin 3 x} {10} + \dotsb} }$

## Proof

By definition of Fourier series:

$\displaystyle \map f x \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$

where for all $n \in \Z_{> 0}$:

 $\ds a_n$ $=$ $\ds \dfrac 1 \pi \int_{-\pi}^\pi \map f x \cos n x \rd x$ $\ds b_n$ $=$ $\ds \dfrac 1 \pi \int_{-\pi}^\pi \map f x \sin n x \rd x$

Thus by definition of $f$:

 $\ds a_0$ $=$ $\ds \frac 1 \pi \int_{-\pi}^\pi \map f x \rd x$ Cosine of Zero is One $\ds$ $=$ $\ds \frac 1 \pi \int_{-\pi}^\pi e^x \rd x$ Definition of $f$ $\ds$ $=$ $\ds \frac 1 \pi \bigintlimits {e^x} {-\pi} \pi$ Primitive of Exponential Function $\ds$ $=$ $\ds \frac 1 \pi \paren {e^\pi - e^{-\pi} }$ $\ds$ $=$ $\ds \frac 2 \pi \dfrac {\paren {e^\pi - e^{-\pi} } } 2$ $\ds$ $=$ $\ds \frac 2 \pi \sinh \pi$ Definition of Hyperbolic Sine

$\Box$

For $n > 0$:

 $\ds a_n$ $=$ $\ds \dfrac 1 \pi \int_{-\pi}^\pi \map f x \cos n x \rd x$ $\ds$ $=$ $\ds \dfrac 1 \pi \int_{-\pi}^\pi e^x \cos n x \rd x$ Definition of $f$ $\ds$ $=$ $\ds \frac 1 \pi \intlimits {\frac {e^x \paren {\cos n x + n \sin n x} } {1 + n^2} } {-\pi} \pi$ Primitive of $e^x \cos n x$ $\ds$ $=$ $\ds \frac 1 \pi \paren {\frac {e^\pi \paren {\cos n \pi + n \sin n \pi} } {1 + n^2} - \frac {e^{-\pi} \paren {\cos n \paren {-\pi} + n \sin n \paren {-\pi} } } {1 + n^2} }$ $\ds$ $=$ $\ds \frac 1 \pi \paren {\frac {e^\pi \cos n \pi} {1 + n^2} - \frac {e^{-\pi} \cos n \paren {-\pi} } {1 + n^2} }$ Sine of Multiple of Pi $\ds$ $=$ $\ds \frac 1 \pi \paren {\frac {e^\pi \paren {-1}^n - e^{-\pi} \paren {-1}^n} {1 + n^2} }$ Cosine of Multiple of Pi $\ds$ $=$ $\ds \frac 2 \pi \frac {\paren {-1}^n} {1 + n^2} \frac {e^\pi - e^{-\pi} } 2$ manipulation $\ds$ $=$ $\ds \frac {2 \paren {-1}^n} {\paren {1 + n^2} \pi} \sinh \pi$ Definition of Hyperbolic Sine

$\Box$

Now for the $\sin n x$ terms:

 $\ds b_n$ $=$ $\ds \frac 1 \pi \int_{-\pi}^\pi \map f x \sin n x \rd x$ $\ds$ $=$ $\ds \frac 1 \pi \int_{-\pi}^\pi e^x \sin n x \rd x$ Definition of $f$ $\ds$ $=$ $\ds \frac 1 \pi \intlimits {\frac {e^x \paren {\sin n x - n \cos n x} } {1 + n^2} } {-\pi} \pi$ Primitive of $e^x \sin n x$ $\ds$ $=$ $\ds \frac 1 \pi \paren {\frac {e^\pi \paren {\sin n \pi - n \cos n \pi} } {1 + n^2} - \frac {e^{-\pi} \paren {\sin n \paren {-\pi} - n \cos n \paren {-\pi} } } {1 + n^2} }$ $\ds$ $=$ $\ds \frac 1 \pi \paren {\frac {-e^\pi n \cos n \pi} {1 + n^2} - \frac {-e^{-\pi} n \cos n \paren {-\pi} } {1 + n^2} }$ Sine of Multiple of Pi $\ds$ $=$ $\ds -\frac 1 \pi \paren {\frac {e^\pi n \paren {-1}^n - e^{-\pi} n \paren {-1}^n} {1 + n^2} }$ Cosine of Multiple of Pi $\ds$ $=$ $\ds -\frac {2 n} \pi \frac {\paren {-1}^n} {1 + n^2} \frac {e^\pi - e^{-\pi} } 2$ manipulation $\ds$ $=$ $\ds -\frac {2 n \paren {-1}^n} {\paren {1 + n^2} \pi} \sinh \pi$ Definition of Hyperbolic Sine

$\Box$

Finally:

 $\ds \map f x$ $\sim$ $\ds \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$ $\ds$ $=$ $\ds \frac 1 2 \frac 2 \pi \sinh \pi + \sum_{n \mathop = 1}^\infty \paren {\frac {2 \paren {-1}^n} {\paren {1 + n^2} \pi} \sinh \pi \cos n x - \frac {2 n \paren {-1}^n} {\paren {1 + n^2} \pi} \sinh \pi \sin n x}$ substituting for $a_0$, $a_n$ and $b_n$ from above $\ds$ $=$ $\ds \frac {\sinh \pi} \pi \paren {1 + 2 \sum_{n \mathop = 1}^\infty \frac {\paren {-1}^n} {1 + n^2} \paren {\cos n x - n \sin n x} }$ simplifying

$\blacksquare$