Euler's Formula

Euler formula connects the complex exponential function with the trigonometric functions. It states that e to the power of i times φ equals the cosine of φ plus i times the sine of φ. Geometrically $e^{(i\varphi{})}$ describes a point on the unit circle in the complex plane whose angle to the real axis is exactly φ. The real part is cos φ, the imaginary part sin φ. The angle φ must be given in radians, not in degrees. This formula is a central tool for representing oscillations, rotations and periodic processes compactly and conveniently for calculation.

If you set $\varphi{} = \pi{}$ in Euler formula, you get $e^{(i\pi{})} = \cos{} \pi{} + i\cdot{}\sin{} \pi{} = -1 + i\cdot{}0 = -1$. Rearranged this gives the famous Euler identity $e^{(i\pi{})} + 1 = 0$. It is considered especially beautiful because it unites five of the most important mathematical constants in a single equation: Euler number e, the imaginary unit i, the circle number π, one and zero. In addition it combines the basic operations of addition, multiplication and exponentiation. It strikingly shows the deep connection between calculus, complex numbers and geometry and is therefore a symbol of mathematical elegance.

Because Euler formula follows from comparing the power series of $e^{x}$, sine and cosine, and these series assume the angle in radians. Only in radians does, for example, the derivative of sin x equal cos x without an extra conversion factor. If you insert the angle in degrees, the series expansions and derivatives no longer match, and the formula gives wrong values. A full turn corresponds to 2π in radians, a half circle to π. Therefore always convert degree values first by multiplying by $\frac{\pi{}}{180}$. In calculus and the complex exponential function, radians are the natural and only correct notion of angle.

Writing Euler formula for +φ and for −φ and combining both lets you isolate sine and cosine. You get $\cos{} \varphi{} =\frac{ (e^{(i\varphi{})} + e^{(-i\varphi{})})}{2}$ and $\sin{} \varphi{} =\frac{ (e^{(i\varphi{})} - e^{(-i\varphi{})})}{(2i)}$. The cosine is therefore the real part, obtained by addition, the sine the imaginary part, obtained by subtraction. These representations are extremely useful because products and powers of trigonometric functions are much easier to compute as exponential expressions. They form the basis of Fourier analysis and the treatment of complex signals in physics and electrical engineering, where oscillations are preferably described in the complex exponential form.

Euler formula makes it possible to describe oscillations and waves in the complex exponential form e^(iωt) instead of with sine and cosine. This greatly simplifies calculations, because differentiating and integrating an exponential function is just multiplication by a factor. In AC circuit theory voltages and currents are represented as complex phasors and one computes with impedances instead of differential equations. In quantum mechanics the wave function contains the factor $e^{(i\varphi{})}$, and in optics interference is described with complex amplitudes. At the end you take the real part to obtain the physically measurable result. Thus the complex notation saves much computational effort.