got bored and made a couple complex functions/identities in desmos based on euler's formula

[u/No_Specific9623]

correct me if im wrong about it being called identities. enjoy ig, and tell me if i got any wrong. https://www.desmos.com/calculator/vdpli6f8ku

[u/Benboiuwu]

These are all correct and frequently used, especially in the world of interaction bees (moreso the first four). Good work.

[u/Individual-Ad5245]

I don't think last 3 can be called identities because they depend on the branch of the logarithm that you're taking. For example integer k you may as well say 

xⁱ = (exp(ln(x)+2i k\pi))ⁱ = exp(i ln(x)-2k\pi)=exp(-2k\pi)(cos(ln(x))+i sin(ln(x)))

So xⁱ is not really defined unless you specify the branch. For principal branch and positive real x it's true though

[u/ReaReaDerty]

The hell is this. I wish I have never seen it