According to the wolfram mathworld, “A Taylor series is a series expansion of a function about a point. A one-dimensional Taylor series is an expansion of a real function about a point is given by

For certain function e^x, sinx, cosx, are like what follows:

Further, e raised by a complex number multiplied by x, will deduce

if x = pie, this leads to the famous Euler formula

Then there are lengthy introduction of complex number, conjugate complex number, on which, all along I am wondering why it’s needed to invent such thing as complex number until

One can see to express a dot formed by x and y, it’s easier to use polar form with the aid of complex unit i, so the product, division of these polar dot can be conducted much efficient way.

Then he went into the harmonic motion problem, the formula can be written as (d^2x/d^2t) = -wt, to solve this linear equation, he’d try “Ansatz”, meaning try and error, using

With the aid of complex number, the solution became

then to

finally get to this succinct format based on Euler’s cos(theta) = (e^io + e^-io)/2

Why the physicists went through such a complex math journey, it’s for real world problem solving.What’s the problem? It’s the harmonic motion, mass spring on a frictional plane problem:

Or, transformed into

According to the math,

Above, we assume the extremity case when r/2 > w0, meaning the friction is hefty enough, while in real work, it’s not for ever oscillation between two ends, nor is it stop rest at center/equilibrium point at first move. It vibrate several times until comes to rest. Reflected on math, it means r/2 < w0.

Next, the problem is driving force exerting on the mass, which is to solve

This is quite hard to solve, so we introduce an artifact image equation, given z = x + i*y (complex number is powerful)