Simple Derivation Of Euler's Formula And Applications

[bhobba]

Here is a simple way to get Euler's relation for those learning pre-calculus/calculus, so the trigonometric addition formula and the derivative of sine and cosine are easy. We will assume some basic knowledge of complex numbers and properties of Euler's number, e.

Consider a small segment of a unit circle with a small angle b at the origin in the complex plane. Draw a line from the ends of the radius of the small arc. You will notice since the angle is small, the line is very close to a tangent of the circle of length b, with it getting better as b gets smaller, i.e. it is at right angles to the radius and of length b. Consider b so small for all practical purposes (don't you love that in non-rigorous calculus) it is at right angles to the radius and length b. Being at right angles means it is multiplied by ib. So take the lower radius of the arc, multiply it by (1+ib), and get the upper radius, i.e. you have rotated it through an angle b. Take any angle b. Divide it by a large number n, and you get (1 +ib/n)^n rotates a complex number through an angle b. You may already know that (1+ ib/n)^n, when n is large, is e^ib, but I will prove it.

d(e^ib)/db = ie^ib or (e^(ib + idb) - e^(ib))/db = ie^(ib). So e^(idb) - 1 = idb. e^(idb) = 1 + idb. Hence e^(ib) = e^(n*ib/n) = (e^(ib/n))^n. If n is large b/n is small so we have e^(ib) = (1+ ib/n)^n.

Hence e^(ib) = e^(ib)*1. This means 1 on the real line of the complex plane is rotated through the angle b. But a complex number of length 1 at an angle a to the real line is cos(a) + i*sine(a).

We have thus shown Eulers famous relation e^(ia) = cos(a) + i*sine(a).

Simply differentiate it and you get ie^(ia) = d(cos(a))/da + i*d(sine(a))/da and we get d(cos(a))/da = sine (a) and d(sine(a))/da = --cos (a).

It is easy to use e^i(a+b) = e^(ia)*e^(ib) to get the formula for sine(a+b) and cos(a+b) and will be left as an exercise for the reader.

Compare how easy deriving these formulas are to the usual way found in textbooks. You can see the real power of complex numbers and why it is so important in mathematics. Textbooks could include these slightly difficult topics that often take pages to prove and make them simple, short and transparent. And the other issue I harp on about - really, people should do calculus and precalculus simultaneously.

BTW this is non-rigorous or intuitive calculus. To make it fully rigorous, you need complex analysis, holomorphic functions and all that. But everyone has to start somewhere.

Thanks
Bill

 

[Office_Shredder]

Very neat.

For what it's worth, you can approximately get the derivatives of cosine and sine just from that triangle you drew at the start for a small rotation.

The line between $(\cos(\theta),sin(\theta))$ and $(\cos(\theta+\epsilon),\sin(\theta+\epsilon))$ is orthogonal to the radius of the circle so is proportional to $(-\sin(\theta),\cos(\theta))$.

 

[scottdave]

This reminds me of the Mathologer video about e^iπ

e to the pi i for dummies

• scottdave
• Apr 16, 2021
• 1
• Complex numbers

This Mathologer video explains e raised to the i pi in a way that even Homer Simpson can understand.