What non-polynomial functions can be "factored"?

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SUMMARY

The discussion centers on the factorization of non-polynomial functions, specifically examining the sine and cosine functions. The user references Euler's proof related to the Basel problem, noting that sin(x) can be expressed as an infinite product involving its roots. The inquiry extends to whether all periodic functions exhibit similar factorization properties, with a specific mention of the cosine function's potential factorization. The Weierstrass factorization theorem is highlighted as a relevant resource for understanding these concepts.

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  • Understanding of infinite series and products
  • Familiarity with periodic functions
  • Knowledge of the Weierstrass factorization theorem
  • Basic trigonometric functions and their properties
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  • Study the Weierstrass factorization theorem in detail
  • Explore the properties of periodic functions and their factorizations
  • Learn about the implications of Euler's work on infinite products
  • Investigate other non-polynomial functions that can be factored
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Mathematicians, students of advanced calculus, and anyone interested in the properties of non-polynomial functions and their factorizations.

MAGNIBORO
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hi, and thanks for come in, sorry for bad english :frown:

I was watching a proof of euler to the basilean problem, and a part of the proof he did this

sin(x) = x ( x + π ) ( x - π ) ( x + 2π ) ( x - 2π ) ( x + 3π ) ( x - 3π) ...

i understand why, but i wanted to know what not polynomial functions they have this property and how
"factored" this functions.

also wanted to know if all periodic functions have this property.

by last
this is true?

cos(x) = ( x + π/2 ) ( x - π/2 ) ( x + π3/2 ) ( x - π3/2 ) ( x + π5/2) ( x - π5/2 ) ...

thanks.
 
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