Current Interruption Transients Calculation, 2nd Edition

Appendix DEuler's Formula

Leonhard Euler (1707–1783) was a prolific Swiss mathematician. His formula¹:

describes the relationship between the oscillatory trigonometric functions, sine and cosine and the complex exponential function where e is the basis of natural logarithms equaling 2.718 281 8 and . e is known as Euler's constant.

The relationship may seem at odds because e^x is a function that increases exponentially with increasing x and sine and cosine functions oscillate repetitively about an axis. Euler discovered the relationship by considering e^x as a power series and Taylor's series for sin x and cos x

Substituting ix for x:

the quantities in brackets being Taylor's series for cos x and sin x, respectively.

Although Euler may not have recognized it, the formula is the basis for complex number theory and links Cartesian and polar coordinates. We can write:

where x is the real part, y is the imaginary part and

Taking the formula as a whole, it represents a unit circle in a ...