Consider a simple trigonometric function
$$ f(x) = \sin cx $$
To compute an approximate value for the function using the second central difference operator (see finite difference methods) we will require $f^{(2)}$ the second partial derivative
$$f^{(2)}(x) \simeq \dfrac{f(x+h) – 2f(x) + f(x-h) }{h^2} = \dfrac{h^2.f^{(2)}(x) + \frac{2}{4!}h^4.f^{(4)}(x)…}{h^2} \tag{1}$$
Now for
\begin{align}
f(x) &= \sin \:cx \; ,\\
f^{(2)}(x) &= – c^2 \sin cx \qquad \text{remember $f’$sin $x =$ cos $x$ and $f’$cos $x =$ -sin $x$}\\
f^{(4)}(x) &= c^4 \sin cx
\end{align}
this strongly suggests that the key part of the second central difference operator may easily be obtained
$$f(x+h) – 2f(x) + f(x-h) = h^2.f^{(2)}(x) + \frac{2}{4!}h^4.f^{(4)}(x)… = \lambda \; \sin \:cx \tag{2}$$
The function to the left spits out a multiple of the original function $f(x)$ and is known as the eigenfunction of the operator. We can prove $(2)$ without the Taylor Series by simply evaluating $\sin(cx + h)$ using known trigonometric properties,
\begin{align}
\sin c(x+h) – 2\sin cx + \sin c(x-h) = & \; \sin cx \cdot \cos ch \; + \; \underbrace{\cos \:cx \cdot \sin ch}_{\downarrow} \\
& -2 \sin cx \\
& + \sin \:cx \cdot \cos ch \; – \; \overbrace{\cos cx \cdot \sin ch}_{\text{(cancel each other)}}\\
= & \; \underbrace{2 (\cos ch \; – \; 1)}_{\lambda} \cdot \underbrace{\sin cx}_{\text{original} \\ \text{function}}
\end{align}
Furthermore, as $\cos 2 \theta = 1 – 2 \sin^2 \theta$,
$$\cos {ch} = \cos 2\dfrac{ch}{2} = 1 – 2\sin^2\dfrac{ch}{2}$$
and
$$2 (\cos ch \; – \; 1) = 2 (- 2\sin^2\dfrac{ch}{2}) = -4\sin^2\dfrac{ch}{2} = \lambda$$
What we have covered so far is closely related to what is known as the discrete Fourier transform also known as Fast Fourier Transform. In a sense when you watch or listen to anything that uses compression technology, things such as video, digital radio, TV, DVD or CD, the mathematics that compresses the data comes out of algorithms built using functions similar to the above.
Next we shall look at the Stability Problems of Explicit Euler