Re: Euler's Formula

Narasimham wrote:
> 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.

It's e^(ix) = (cost x + i sin x)

Does e^(ix) = (sin x + i cos x) have any solutions? I guess only where
cos x + i sin x = sin x + i cos x.

e's a constant. e^ix is a constant exponential function with one free
variable.

Ross

.

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