/sci/ - Science & Math » Thread #14129712

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Anonymous Thu 20 Jan 23:05:47 2022 No.14129712 View Reply Original Report

Quoted By: >>14129730 >>14131065 >>14131085 >>14131089

>e^i(pi)=-1
>(-1)^i(pi)=-1
>0^i(pi)=undefined
does e = -1?

Anonymous

Anonymous Thu 20 Jan 2022 23:11:33 No.14129730 Report

Quoted By: >>14129736 >>14131059 >>14131089

>>14129712
No. e^{ix) is multi-valued.

Anonymous

Anonymous Thu 20 Jan 2022 23:13:41 No.14129736 Report

Quoted By: >>14131085 >>14131089

>>14129730
How does that work?

Anonymous

Anonymous Fri 21 Jan 2022 08:40:31 No.14131059 Report

Quoted By:

>>14129730
Wrong

Anonymous

Anonymous Fri 21 Jan 2022 08:46:25 No.14131065 Report

Quoted By:

>>14129712
>a^0 = 1
>b^0 = 1
>0^0=undefined
does a = b?

Anonymous

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Anonymous Fri 21 Jan 2022 09:01:16 No.14131085 Report

Quoted By: >>14131097

>>14129712
>>14129736
Now e^x=e^y does imply x=y when x,y are real but the exponential function is multivalued given a complex argument. Think of the sine function:
sin(0)=sin(2pi), this however doesn't imply that 0=2pi, same story here.
f(x)=f(y) implies x=y only when f is one to one.

Anonymous

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Anonymous Fri 21 Jan 2022 09:04:53 No.14131089 Report

Quoted By:

>>14129712
but that's wrong
$(-1)^{i\pi}=e^{\ln(-1^{i\pi})}=e^{i\pi\ln(-1)}=e^{(i\pi)^2}=e^{-\pi^2}\approx0.0000517$
where ln is the complex logarithm. wolfram alpha is your friend.
>does e = -1
of course not!
>>14129736
>>14129730
e^{ix} is not multivalued, but the complex logarithm is. this is why the principle value is usually taken by default.
>how does that work
It's just like how "arctan(15)" technically gives you an infinite number of angles, all differing by 1 rotation or how "square root of 9" can feasibly include 3 and -3. any function that is not one-to-one, like x^2 or tan(x) or exp(ix), produces a multivalued inverse.

Anonymous

Anonymous Fri 21 Jan 2022 09:12:39 No.14131097 Report

Quoted By: >>14131137

>>14131085
>the exponential function is multivalued
wrong

Anonymous

Anonymous Fri 21 Jan 2022 09:37:33 No.14131137 Report

Quoted By: >>14131186

>>14131097
Many-to-one, not multivalued, my bad
Point still stands

Anonymous

Anonymous Fri 21 Jan 2022 10:04:37 No.14131186 Report

Quoted By: >>14131479

>>14131137
literally what?
"complex" is not multivalued

Anonymous

Anonymous Fri 21 Jan 2022 12:29:37 No.14131479 Report

Quoted By:

>>14131186
I already said it's not multivalued (one to many), it's instead many to one:
The complex exponential function is many-to-one, meaning that more than one input (specifically those that differ by 2?i) can yield the same output.
Not sure what's so hard to understand.

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