Because Fourier series in two and three dimensions are typically written as a sum of complex exponentials, some properties of exponential functions are reviewed here. The derivative of an exponential function is an exponential function.

$$\dfrac{d}{dx}e^x = e^x.$$

The product of two exponential functions can be written as one exponential function by adding the exponents,

$$e^ae^b=e^{a+b}. $$

Complex exponentials are related to circular functions by Euler's formula,

$$e^{ix}=\cos x + i\sin x.$$

Using the product rule, $e$ raised to an arbitrary complex number $a +ib$ can be written as,

$$e^{a+ib}=e^a\left(\cos b + i\sin b\right).$$

An important application of exponential functions is the solution to all linear differential equations. For example, the equation for a damped mass-spring system is,

$$F=m\dfrac{d^2x}{dt^2}= -b\dfrac{dx}{dt}-kx,$$

where $m$ is the mass, $a = \dfrac{d^2x}{dt^2}$ is the acceleration of the mass, $F = ma$ is the total force, $-b\dfrac{dx}{dt}$ is a damping force that is in the opposite direction as the velocity $\dfrac{dx}{dt}$, and $-kx$ is the spring force that is in the opposite direction as the extension of the spring.

For $m = 0.5$, $b=1$, $k=1$, the equation is,

$$0.5\dfrac{d^2x}{dt^2}+\dfrac{dx}{dt}+x=0.$$

Substitute $x=e^{\lambda t}$. Because the derivative of an exponential function is proportional to the derivative of an exponential function, all of the terms are proportional to $e^{\lambda t}$,

$$0.5\lambda^2 e^{\lambda t}+\lambda e^{\lambda t}+ e^{\lambda t}=0.$$

The exponential factors cancel out,

$$0.5\lambda^2 \require{cancel}\cancel{e^{\lambda t}}+\lambda \require{cancel}\cancel{e^{\lambda t}}+ \require{cancel}\cancel{e^{\lambda t}}=0,$$

resulting in a quadratic equation for $\lambda$.

$$0.5\lambda^2+\lambda +1 =0.$$

This can be solved with the quadratic equation,

$$\lambda_{\pm} = -1\pm i$$

The two solutions to the differential equation are,

$$x_+(t) = e^{\lambda_+t}=e^{-t}\left(\cos(t)+i\sin(t)\right), \\ x_-(t) = e^{\lambda_-t}=e^{-t}\left(\cos(t)-i\sin(t)\right)$$

Any solution can be expressed as a linear combination of these two solutions,

$$x(t) = C_+x_+ + C_-x_-.$$

Most of the time, we seek real solutions. If $ C_+ = C_-$ is real, the $\sin$ terms cancel, and the solution is real. If $ C_+ = -C_-$ is imaginary, the $\cos$ terms cancel and the solution is real. In general, the solution will be real if,

$$C_- = C_+^*.$$