Odd/even for a multivariable function

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SUMMARY

The discussion clarifies the definition of odd and even functions in multiple variables, confirming that the standard definition is f(-x,-y) = -f(x,y). It highlights that functions can be categorized as odd or even based on specific sets of variables, particularly when involving parameters. For instance, a family of even functions of x and y, parametrized by z, can be expressed as F(x,y;z), where F(-x,-y;z) = F(x,y;z) holds true for all values of z. The distinction between being even in all variables versus specific variables is emphasized.

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  • Investigate the implications of function parametrization on symmetry
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Is the definition of an odd/even function in multiple variables what I would expect it to be, ie

f(-x,-y)=-f(x,y)

Thanks!
 
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yes that is the standard definition. Sometimes I have a family of odd or even functions however, i.e. the functions involve two kinds of variables, the variables of the functions, and the variables that parametrize the family. Then they are only odd or even in the first set of variables.e.g. a family of even functions of x,y, parametrized by z, might be represented as a single function F(x,y;z) such that for all x,y,z we have F(-x,-y;z) = F(x,y;z).i.e. for each value of z, say z=c, the function F(x,y;c) is even in (x,y).

But if you just say "even", rather than "even in (x,y)", then yes I would expect it to be even in all variables present.
 
awesome thank you!
 

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