Real line : Complex plane :: Hat : ?

The title is a word analogy puzzle. The plots below are hints. In each pair, the black curve is the same.

Arctangent

Arctangent on the real line
On the real line
Arctangent on the complex plane
On the complex plane

Exponential function

Exponential function on the real line
On the real line
Exponential function on the complex plane
On the complex plane

Sine function

Sine function on the real line
On the real line
Sine function on the complex plane
On the complex plane

x4

x^4 on the real line
On the real line
x^4 on the complex plane
On the complex plane

Answer: boa constrictor digesting an elephant.

3 thoughts on “Real line : Complex plane :: Hat : ?”

  1. “A picture is worth a thousand words”, but in this case it’s not true.
    It would be essential to mention that each surface is the graph of the real part of the corresponding complex function.

  2. I guess it is not obvious to me what the analytic continuation of the “hat graph” should be, but I did like the Little Prince reference.

  3. My analysis prof sketched this on the board when I was an undergrad: imgur dot com slash eF6OzCc. I was the only person in the classroom chuckling.

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