Re: Euler's Formula

Ross A. Finlayson wrote:
> 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) = (cos 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.

As already stated it's an identity, an equation valid for all values of
the variable x.

.

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  • Re: Eulers Formula
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  • Re: Eulers Formula
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