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Let $D$ be a nonempty open subset of $\mathbb{R}\times\mathbb{R}$ and $f:D\to\mathbb{R}$ be a function of two variables. For all $(x,y)\in D$ and $t>0$ such that $(tx,ty)\in D$, if the equality $f(tx,ty)=t^\alpha f(x,y)$ is valid for some real number $\alpha\in\mathbb{R}$, then we call $f$ a homogeneous function of order $\alpha$ on $D$.

Could you please recommend literature containing basic knowledge of the above-defined homogeneous functions for me to cite in a manuscript? Thank you very much!

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The most convenient source of information on homogeneous functions is Wikipedia; if that is not satisfactory as a source, you could cite:

J.P. Lewis, Homogeneous Functions and Euler’s Theorem. In: An Introduction to Mathematics (Palgrave Macmillan, London, 1969).

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    $\begingroup$ Thank you very much for your answer, Dr. Carlo Beenakker $\endgroup$
    – qifeng618
    Commented Jun 27 at 12:06
  • $\begingroup$ Is the following baisc property contains in the reference by Lewis? Let $f:D\to\mathbb{R}$ be a differentiable homogeneous function of order $n$. Then $\partial_1f$ and $\partial_2f$ are homogeneous functions of order $n-1$ and $$ x\partial_1f(x,y)+y\partial_2f(x,y)=n f(x,y). $$ In particular, if $n=1$ and $f$ is twice differentiable on $D$, then \begin{align*} x\partial_1f(x,y)+y\partial_2f(x,y) &=f(x,y),\\ x\partial_1^2f(x,y)+y\partial_1\partial_2f(x,y) &=0,\\ x\partial_1\partial_2f(x,y)+y\partial_2^2f(x,y) &=0. \end{align*} $\endgroup$
    – qifeng618
    Commented Jun 27 at 13:46
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    $\begingroup$ I will email you a copy. $\endgroup$
    – Carlo Beenakker
    Commented Jun 27 at 13:51
  • $\begingroup$ I received your e-mail. Thank you a lot. $\endgroup$
    – qifeng618
    Commented Jun 27 at 13:52
  • $\begingroup$ The above differential properties are contained in the paper by Lewis. But I don't find the following integral property in it: Let $f:D\to\mathbb{R}$ be a homogeneous function of order $n$ and let $[0,x]\times [0,y]\subseteq D$. If $u\mapsto f(u,y)$ is integrable on $[0,x]$ (or $v\mapsto f(x,v)$ is integrable on $[0,y]$, respectively), then the function $x\mapsto\int_0^{x}f(u,y)\operatorname{d}\!u$ (or the function $x\mapsto\int_0^{y}f(x,v)\operatorname{d}\!v$, respectively) is a homogeneous function of order $n+1$. $\endgroup$
    – qifeng618
    Commented Jun 27 at 14:16

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