The omnipresent Google informed me today about Euler’s birthday. I responded by trying to recall how the famous Euler equation is derived. Being always a lousy mathematician (me, not Euler), I had to peek into my college books. What I found must I write in hope that I will not forget anytime soon.

So… for real numbers, the exponential function can be represented by the following infinite series:

${e }^{a }=\sum _{k =0 }^{\infty }\frac{{a }^{k }}{k !}\;$

Therefore, it is not unreasonable to define the exponential function for complex numbers the equivalent way:

${e }^{z }=\sum _{k =0 }^{\infty }\frac{{z }^{k }}{k !}\;$

As many know it, in the real domain the most basic property of exponential functions is that ${e }^{a +b }={e }^{a }{e }^{b }$.  It is also possible to prove that for the above-defined complex version of the exponential function the equivalent claim is true: ${e }^{z _{1 }+z _{2 }}={e }^{z _{1 }}{e }^{z _{2 }}$. However, the proof that

${e }^{z _{1 }+z _{2 }}=\sum _{k =0 }^{\infty }\frac{{{\left( z _{1 }+z _{2 }\right) }}^{k }}{k !}\; ={\left[ \sum _{k =0 }^{\infty }\frac{{z _{1 }}^{k }}{k !}\; \right] }{\left[ \sum _{k =0 }^{\infty }\frac{{z _{2 }}^{k }}{k !}\; \right] }$

is just too intimidating to me. As always in similar situation, I am eager to assume that the claim ${e }^{z _{1 }+z _{2 }}={e }^{z _{1 }}{e }^{z _{2 }}$ is correct because that’s what people are telling me. You see, regarding mathematics I am mostly a believer.

Okay, by avoiding the proof elegantly, I am free to continue… A less general clam must therefore also be true:

${e }^{z }={e }^{x +i y }={e }^{x }{e }^{i y }$

Where x and y are real numbers. Therefore, one can write:

${e }^{x +i y }={e }^{x }{e }^{i y }=$

$={e }^{x }{\left( 1 +\frac{y }{1 !}i -\frac{{y }^{2 }}{2 !}-\frac{{y }^{3 }}{3 !}i +\frac{{y }^{4 }}{4 !}+\frac{{y }^{5 }}{5 !}i -\dots \right) }$

$={e }^{x }{\left[ {\left( 1 -\frac{{y }^{2 }}{2 !}+\frac{{y }^{4 }}{4 !}-\dots \right) }+i {\left( \frac{y }{1 !}-\frac{{y }^{3 }}{3 !}+\frac{{y }^{5 }}{5 !}-\dots \right) }\right] }$

Because Euler was never as blind as I am, regarding mathematics at least, he probably was quick to realize that the first sum of real numbers represents the cosine function and the second sum of real numbers represents the sine function. Therefore:

${e }^{z }={e }^{x +i y }={e }^{x }{\left[ \cos y + i \sin y\right] }$

Or, if we suppose a less general case of purely imaginary z:

${e }^{i \phi }=\cos \phi + i \sin \phi$

Or, in even less general case of $\phi =\pi$, we have:

${e }^{i \pi }=-1$

A personal note… I was a very relaxed student, never taking too much attention during lectures. This particular day on the college math class I was lost as usual. Our professor was doing some heavy math on the blackboard but my attention drifted away sometime at the very beginning. I was daydreaming about who knows what – possibly about college girls – and was only marginally aware of some math at the blackboard. Several seconds before the class end, our professor encircled this small equation. WTF!? Did he just throw a hammer into my forehead? Other people left the room, but I remained sitting there, looking at the blackboard, trying to figure out what went wrong. How in the world you can put the $e$ and the $\pi$ into the same, ultrasimple equation?

Advertisements