Re: Euler's Formula

Anand P. Paralkar wrote:

> Is there any "intuitive" way to understand Euler's Formula:
> e^(ix) = cosx + i sinx ; Imagining e^ix, based on what e^x is, is impossible.

e^(ix) consists of sin,cos oscillating functions,so you need to
consider a second order generating differential equation say y '' + y =
0, y = A cos(x) + B sin(x) that should be manipulated to include first
order y ' + y = 0, y = y0 * e^x. I let some free thought,albeit vague
along lines you suggested, any comments are welcome.

A and B may be real, or complex constants. To reduce to a single
constant we may choose the signs of A^2 + B^2 without loss of
generality for this purpose.

If A^2 + B^2 > 0 , y = ymax* cos(x +/- a) ; a is phase difference of a
real sine-wave.

( If A^2 - B^2 = 0 , y = ymax * sin (x +/- pi/4) is a special case of
the above).

If A^2 + B^2 = 0 y = ymax * e ^(+/- i x) .. is the complex Euler
identity wave function you are perhaps trying to connect e^ix to e^x
intuitively as an engineer. Elliptic type.

If A^2 + B^2 < 0 , y = ymax* ( tan(a)*sin(x) (+/-) i sec(a)*cos(x) )is
a Hyperbolic type complex wave function.I wonder why this has not been
an identity function akin to Euler's.

.

Relevant Pages

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  • Re: Eulers Formula
    ... >> e^consists of sin,cos oscillating functions,so you need to ... >> real sine-wave. ... >> a Hyperbolic type complex wave function.I wonder why this has not been ...
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  • Re: Eulers Formula
    ... > e^consists of sin,cos oscillating functions,so you need to ... > real sine-wave. ... > a Hyperbolic type complex wave function.I wonder why this has not been ...
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