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
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)
Then we get