Euler's Relation Explained - No Derivation Needed

[ragavcit]

Hello everyone,
Can anyone give me a brief explanation about how EULER derived the formula exp(j*)=cos*+jsin* without going into his actual derivation..

 

[nicksauce]

I'm not sure exactly how Euler did it, but a simple way to do it is to write out the taylor series for both sides of the equation and showing that they are equal.

 

[lurflurf]

I think Euler used series expansion. This result is not really derived though, since it is the extension of the definition into a larger domain. Usually several properties of
exp:R->R
are chosen to be preserved in the extension
exp:C->C
and the formula folows naturally
I like preserving these properties
exp(a+b)=exp(a)*exp(b)
and
lim_{x->0}[exp(x)-1]/x=1

 

[yasiru89]

Euler did use the series expansion, plugging in [itex]\theta := i\theta[/tex] in the defining series for the exponential,

$$\exp{\theta} = \sum_{n = 0}^{+\infty} \frac{\theta^{n}}{n!}$$

and seperating real and imaginary parts. It was a phenomenal achievement in the true Eulerian spirit. A bit of a shame that people overlook this 'derivation' entirely and choose to define the exponential from scratch nowadays.

 

[lurflurf]

A bit of a shame that people overlook this 'derivation' entirely and choose to define the exponential from scratch nowadays.

It is overlooked because it is not a 'derivation'. You cannot 'derive' an identity of a function without first giving a definition of the function. It is possible to extend the real exponential to the complex plane by analytic continuation, and this is done in introductory books with titles like analysis or complex variables. The approach involves Heavy lifting.

 

[yasiru89]

It is overlooked because it is not a 'derivation'. You cannot 'derive' an identity of a function without first giving a definition of the function. It is possible to extend the real exponential to the complex plane by analytic continuation, and this is done in introductory books with titles like analysis or complex variables. The approach involves Heavy lifting.

This is precisely why I chose to emphasise 'derivation', however, my criticism is not of the analytic extension as such, but of the practice to simply define,

$$\exp{it} = \cos{t} + i\sin{t}$$.

A better approach is to define the complex analogue as,

$$\exp{z} = \sum_{n = 0}^{+\infty} \frac{z^{n}}{n!}$$
, establish global convergence and proceed to show as Euler did that this may be considered to be composed of two components which in themselves may be used to define (or show that they are the corresponding Taylor expansions about the origin of) the cosine and sine functions.