@Reconstructo,
Reconstructo;153830 wrote:Here you go. I think a series is the only way for a case like this.
Here's one way of making Euler's identity plausible without using series, or at least without using them directly.
(It's based on an impromptu post I made to sci.math on 19 May 2008, but I think it holds up OK.)
The act of faith required is to believe that: (i) there exists a differential calculus of complex-valued functions of complex variables; (ii) there exists a complex-valued generalisation of the exponential function, exp; (iii) even when understood as a complex-valued function, exp is equal to its own derivative.
I'll write exp(z) as e^z (with the understanding that the power notation for complex numbers is not well-defined in general).
Consider u(t) = e^{it}, as a complex-valued function of the real variable t.
Given the above acts of faith (which can all be justified, of course), it is reasonable to guess, correctly, that u satisfies the differential equation:
u'(t) = iu(t)
with the initial condition:
u(0) = 1.
If you are happy with the idea that multiplication by i means turning through a right angle anticlockwise in the complex plane, then this means that if u(t) is considered as a vector in the complex plane, its derivative is the same vector turned through an anticlockwise right angle.
But this is precisely the condition satisfied by the point u(t) on a circle of unit radius, centre the origin, at arc length t anticlockwise from the point (1, 0), which is as usual identified with the number 1 on the real axis.
(You can picture it: an infinitesimally small change in u(t) is almost exactly equal in length to the infinitesimal increment in the arc length t, and is tangential to the circle, therefore the derivative u'(t) is the unit vector tangent to the circle in the anticlockwise direction at the point u(t). Ignore Bishop Berkeley turning in his grave - we're talking acts of faith here!)
So it is reasonable to suppose that this gives the solution of the differential equation.
Therefore, u(pi) should be the point on the unit circle at arc length pi anticlockwise from the point (1, 0), i.e. it is the point (-1, 0), which is identified with the number -1 on the real axis.
Hence (by this plausible hand-waving argument):
e^{i*pi} = -1.
Series might have to be used in the rigorous justification of this hand-waving argument, but I hope that that that doesn't take away all of its charm! The full original thread in sci.math is here:
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