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I am working with hyperbolic functions and was wondering where they actually came from. I am under the understanding that Ricotta (and I think Johann Heinrich Lambert also did work in this area), did major work in this area however I am wondering how he got to the final definition of sinh and cosh using the exponential function.

Did Ricotta or Lambert ever publish a research paper which I can access somehow? I really want to understand where they come from (or if someone can show me).

Another question I had was why people required finding the Taylor expansion for both sing and cosh, what is the purpose? Why is it useful?

Thank you so much,

user2250537
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  • This masters thesis with a bit of history http://scimath.unl.edu/MIM/files/MATExamFiles/Schutz_MATpaper_FINAL.pdf can be a good starting point. – Martín-Blas Pérez Pinilla Dec 05 '14 at 09:12
  • You should ask your second question separately, you'll get better answers. –  Dec 05 '14 at 09:16
  • I have no insight into the historical matters, but the connection to the circular functions with imaginary arguments seems obvious. $\cos ix=(e^{i^2t}+e^{-i^2t})/2=\cosh t$, $\sin it=(e^{i^2t}-e^{-i^2t})/2i=i\sinh t$, and the unit circle turns to the hyperbola $x^2-y^2=1$. –  Dec 05 '14 at 09:18
  • Ok thank you guys so much! – user2250537 Dec 05 '14 at 09:27

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A rather complete history in Enter, Stage Center: The Early Drama of the Hyperbolic Functions.

Martín-Blas Pérez Pinilla
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