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I'm interested, why are $\cosh$ and $\cos$ related by the imaginary unit like$$\cos (x)=\cosh (ix)?$$Is there any visual proof? How can be circle related to hyperbola by complex unit?

Andrew D. Hwang
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Adam Červenka
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  • This is by definition, and by intent. For $$x \in \Bbb{R}, ~: \cos(x) = \frac{e^{ix} + e^{-ix}}{2}.$$ Also, $\cosh(x)$ is defined to be $$\frac{e^{x} + e^{-x}}{2}.$$ Therefore, $$\cosh(ix) = \frac{e^{ix} + e^{-ix}}{2} = \cos(x).$$ – user2661923 May 28 '22 at 18:44
  • This is not exactly what I wanted, so edited question. How can be something derived from circle changed by complex unit to make hyperbola? – Adam Červenka May 28 '22 at 18:54
  • It seems as if you are attempting to raise an interesting question. Perhaps other reviewers will be able to perceive what you are trying to ask with respect to your references to a circle and a hyperbola. Personally, I do not understand what you are trying to ask. If no one else provides a helpful response to your posting, as is, then I will need you to be much clearer on what you are trying to ask. – user2661923 May 28 '22 at 18:57
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    @Adam How do you define $\cos x$ and $\cosh x $… that is the question? – mathcounterexamples.net May 28 '22 at 19:00
  • @mathcounterexamples.net For his posting as is, except for the circle <--> hyperbola reference that I don't understand, then yes, that is the question. However, my previous comment covered that, and the OP (i.e. original poster) indicated that he has a different question in mind. The difficulty is, until he states his alternative question in much clearer detail, all that one can do is fall back on the definitions. – user2661923 May 28 '22 at 19:10
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    Maybe the better question would be is there any transformation that transform circle $x^2 + y^2 = 1$ into hyperbola $x^2 - y^2 = 1$? – Adam Červenka May 28 '22 at 19:18
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    Could you give me an example, in general, of what you have in mind by the term transformation? Also, explanatory information belongs in the edited posting, not the comments. Also, if you want a specific person to be flagged with a comment, you have to include the @... flag. – user2661923 May 28 '22 at 19:20
  • Something like linear transformation or Legendre transformation or involution - something that "stretches" circle to hyperbola. – Adam Červenka May 28 '22 at 19:32
  • I'm working on visualisation of "complex" rotation (rotation by complex angle). – Adam Červenka May 28 '22 at 19:35
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    @Adam Červenka: Just map $y\leftrightarrow iy$ will do. – emacs drives me nuts May 28 '22 at 19:36
  • Okey... So my last question when I rotate point $(0,1)$ by angle $\theta + i \phi, \theta, \phi \in R$, how would that point looks like after rotation? – Adam Červenka May 28 '22 at 20:02
  • In https://math.stackexchange.com/a/455625/35416 I'm relating the imaginary unit to alternating signs in the series expansion. It's not visual but perhaps it can be of some use. For rotation by a complex angle you would likely need 4d space to visualize this, since each complex-valued dimension would require 2 real-valued ones to depict. Perhaps you can find a nice projection to make that "only" 3d. – MvG May 29 '22 at 19:45

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