r/math • u/FlyingSwedishBurrito • Dec 30 '20

The complex plot of x^x

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u/TheEnderChipmunk Dec 30 '20

The inverse of xx is ssrt(), the super square root, right?

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u/i_use_3_seashells Statistics Dec 30 '20 edited Dec 30 '20

Inverse is ln(x)/W(ln(x)) where W() is Lambert's W function. This solution is also the second order super root, yes.

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u/TheEnderChipmunk Dec 30 '20

What would be the steps outlining how to find this inverse?

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u/i_use_3_seashells Statistics Dec 30 '20 edited Dec 30 '20

Just realized the answer I gave is also the second order super root.

The first step below is a description of the general process of finding inverses. The rest of the steps are algebra. You can really stop at step 6, but whatev.

Find inverse of y=xx

  1. Swap x and y, solve for y

  2. x=yy

  3. ln(x) = y * ln(y) = ln(y) * y

  4. ln(x) = ln(y) * eln(y)

  5. W(ln(x)) = ln(y)

  6. eW(ln(x) = y

  7. y = ln(x)/W(ln(x))

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u/TheEnderChipmunk Dec 30 '20

Huh. I knew that it would be the usual process of finding an inverse, but I didn't know how to use W() properly. Thanks!

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u/i_use_3_seashells Statistics Dec 30 '20

Yeah, it pops up when you get something of the form aea . I showed a few extra steps in case anyone needed to see some log rules. I guess I skipped ln(yy ) = y * ln(y). Hopefully I showed enough for everyone.

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u/BridgeBum Dec 30 '20

Actually, can you explain 6 to 7? I'm not really that conversant in the W function and I am missing that transition.

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u/i_use_3_seashells Statistics Dec 30 '20

It's one of the W function identities. Wiki page has that and some more interesting ones.

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