Homogeneous Function of Degree Example

Define Homogeneous Function with Example

A function f defined by

u\;=\;f\;(x,\;y,\;z,\;…)

of any number of variables is said to be homogeneous of degree n in these variables if multiplication of these variables by any number t(\neq0) result in the multiplication of the function by t^n, i.e.,

f\;(tx,\;ty,\;tz,\;…) = t^n f\;(x,\;y,\;z,\;…) (1)

provided that (tx,\;ty,\;tz,\;…) is in the domain of f.

Taking t=\frac1x,(x\neq0) the equation (1) becomes

  • f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;=\;\frac1{x^n}f(x,\;y,\;z,\;…)\;
  • \;f(x,\;y,\;z,\;…)\;\;=x^n\;f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;
  • =x^ng\left(\frac yx,\;\frac zx\;…\right)

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