Behaviour of z^Infinity and Infinity^z

[harrysarson]

What does the Riemann mapping say when it comes to the complex exponential?

@derknorton
@balagge

[derknorton]

My understanding is that all mathematical operations that work in the complex plane work the same way on the Riemann sphere. So the results of complex exponential operations that land in the complex plane get mapped to the appropriate place on the Riemann sphere. This includes zero and all four infinities (which map to the poles).

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On Mar 9, 2018, at 5:36 AM, Harry Sarson ***@***.***> wrote: What does the Riemann mapping say when it comes to the complex exponential? @derknorton <https://github.com/derknorton> @balagge <https://github.com/balagge> — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub <#24>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AFk1eYg5265IBEFelQUwroPF1Ie1_shAks5tcndUgaJpZM4SkNEU>.

[harrysarson]

What does that make Complex(Infinity).exp() equal to?

[derknorton]

Since z^n is the same as z * z * ... * z and Infinity * Infinity => Infinity I would say the answer should be Infinity.

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On Mar 9, 2018, at 8:46 AM, Harry Sarson ***@***.***> wrote: What does that make Complex(Infinity).exp() equal to? — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub <#24 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AFk1eWHCXBVLQqHp6JEpvCkubJCgNAeGks5tcqPggaJpZM4SkNEU>.

[balagge]

indeterminate. (no value, NaN, whatever. same as 0 / 0)

[balagge]

wolfram:

[balagge]

This is because

which means that if z = x + iy and z goes to infinity, then we cannot determine the limit of exp(z). For example, take the following z values: 0, i, 2i, 3i, .... Then the limit exp(0), exp(i), exp(2i), ... will be meaningless, as exp(n i) = cos n + i sin n, which is essentially a "random" point on the unit circle, clearly does not converge (neither to infinity, nor to any given complex number)

[balagge]

or, if you take the z values: 0, i*pi, 2 i*pi, 3 i*pi, etc. (clearly z goes to infinity) then the exp(z) values will be +1, -1, +1, ... as exp(n i pi) is -1 if n is odd and + 1 if n is even.
complex exponentiation is always tricky.

[balagge]

truly confusing, I never liked this part of complex theory.
https://en.wikipedia.org/wiki/Exponential_function#Complex_plane

[balagge]

this is how wikipedia pictures the complex exp function. pretty cool. Brightness means the absolute value, color means the direction.

Now if you approach infinity "to the right", then you will have a sequence that also goes to infinity. However, if you approach it "to the top" (as in the example I wrote above), then you will be going around the unit circle. Same with the bottom direction.

If you go left (to the same complex infinity) than the exponentiation will go to zero...

https://upload.wikimedia.org/wikipedia/commons/a/a7/Complex_exp.jpg

[derknorton]

I think Harry was asking about Infinity^z rather than z^Infinity. The first should be Infinity. I think you are correct that the second should be indeterminate (NaN).

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On Mar 9, 2018, at 9:27 AM, Paál Balázs ***@***.***> wrote: wolfram: <https://user-images.githubusercontent.com/13258533/37217878-1069145a-23bf-11e8-92be-e8594f029ba3.png> — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub <#24 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AFk1ec-FmRT0CCgVU8626ah22wFIN2FZks5tcq1ygaJpZM4SkNEU>.

[harrysarson]

@derknorton Both cases are relivant but to clarify:

Complex(Infinity).exp() calculates e^Infinity so I was talking about the second case.

[derknorton]

Sorry, my mistake...

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On Mar 9, 2018, at 11:29 AM, Harry Sarson ***@***.***> wrote: @derknorton <https://github.com/derknorton> Both cases are relivant but to clarify: Complex(Infinity).exp() calculates e^Infinity so I was talking about the second case. — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub <#24 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AFk1eZLnTvBQSoALVpXUiMjs0enyuLT7ks5tcsoQgaJpZM4SkNEU>.

[harrysarson]

To summarise for complex z :

1. 0 ^ Infinity === 0
2. 1 ^ Infinity === 1
3. z ^ Infinity === |z| < 1 ? 0 : NaN
4. Infinity ^ Infinity === NaN
5. Infinity ^ 0 === NaN
6. Infinity ^ 1 === Infinity
7. Infinity ^ z === Infinity

maybe?

[derknorton]

I think #3 should result in Infinity for |z| > 1

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On Mar 9, 2018, at 11:36 AM, Harry Sarson ***@***.***> wrote: To summarise for complex z : 0 ^ Infinity === 0 1 ^ Infinity === 1 z ^ Infinity === |z| < 1 ? 0 : NaN Infinity ^ Infinity === NaN Infinity ^ 0 === NaN Infinity ^ 1 === Infinity Infinity ^ z === Infinity — You are receiving this because you were mentioned. Reply to this email directly, view it on GitHub <#24 (comment)>, or mute the thread <https://github.com/notifications/unsubscribe-auth/AFk1eVGBKrsd7RXaE1CKXj1aXgiOwA_Cks5tcsusgaJpZM4SkNEU>.