Euler's Formula

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z4lis
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Euler's Formula

Postby z4lis » Mon Jul 07, 2008 5:52 pm UTC

e's reign over exponents is rather unfair, I think it's time to let all those other numbers have a chance in the spotlight!

aix = cos(x*ln(a)) + i*sin(x*ln(a))

The odd thing is... I've never seen that above equation, until I derived it last night. Surely it's been floating around for years, and is not widely known because e is an attention whore? :P
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SimonM
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Re: Euler's Formula

Postby SimonM » Mon Jul 07, 2008 6:02 pm UTC

z4lis wrote:e's reign over exponents is rather unfair, I think it's time to let all those other numbers have a chance in the spotlight!

aix = cos(x*ln(a)) + i*sin(x*ln(a))

The odd thing is... I've never seen that above equation, until I derived it last night. Surely it's been floating around for years, and is not widely known because e is an attention whore? :P


Its a trivial consequence of Euler's formula

aix = eln(a)ix

Which is probably why no one bothers with it
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NathanielJ
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Re: Euler's Formula

Postby NathanielJ » Mon Jul 07, 2008 6:05 pm UTC

SimonM wrote:Its a trivial consequence of Euler's formula

aix = eln(a)ix

Which is probably why no one bothers with it


Similarly, the derivative of ax (w.r.t. x) is ln(a)*ax, which can be derived easily from the (simpler) derivative rule for ex, but most people don't bother remembering this rule. Hence why the rules that involve e are usually the only ones people bother remembering.
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z4lis
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Re: Euler's Formula

Postby z4lis » Mon Jul 07, 2008 6:24 pm UTC

I realize it's trivial, which is exactly why I'm curious as to why we don't use the more general form. It seems sort of bizarre to me.
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SimonM
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Re: Euler's Formula

Postby SimonM » Mon Jul 07, 2008 6:35 pm UTC

z4lis wrote:I realize it's trivial, which is exactly why I'm curious as to why we don't use the more general form. It seems sort of bizarre to me.


The simple form has more uses in the area of complex numbers, ie rotation angle A is multiplication by e^{iA}
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Re: Euler's Formula

Postby ossix » Mon Jul 07, 2008 6:36 pm UTC

z4lis wrote:I realize it's trivial, which is exactly why I'm curious as to why we don't use the more general form. It seems sort of bizarre to me.


because your "more general form" uses ln() - which is the logarithm to the base e, therefore being not a bit "more general". ;-)

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z4lis
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Re: Euler's Formula

Postby z4lis » Mon Jul 07, 2008 6:58 pm UTC

SimonM wrote:
z4lis wrote:I realize it's trivial, which is exactly why I'm curious as to why we don't use the more general form. It seems sort of bizarre to me.


The simple form has more uses in the area of complex numbers, ie rotation angle A is multiplication by e^{iA}


Doesn't the other form have the same property. It's essentially the same thing, but when the base is not e, we have faster or slower "rotation".
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SimonM
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Re: Euler's Formula

Postby SimonM » Mon Jul 07, 2008 7:04 pm UTC

No, it doesn't rotate by x, but by ln a x
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Re: Euler's Formula

Postby Robin S » Mon Jul 07, 2008 7:04 pm UTC

Your argument strikes me as similar to saying "why multiply by n when we can add the number to itself n times?" It's needlessly complicating matters without making anything clearer, simpler or easier.
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