Timeline for Literature containing basic knowledge of homogeneous functions
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 29 at 0:56 | vote | accept | qifeng618 | ||
| Jun 27 at 14:16 | comment | added | qifeng618 | 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 $n+1$. | |
| Jun 27 at 13:52 | comment | added | qifeng618 | I received your e-mail. Thank you a lot. | |
| Jun 27 at 13:51 | comment | added | Carlo Beenakker | I will email you a copy. | |
| Jun 27 at 13:46 | comment | added | qifeng618 | 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*} | |
| Jun 27 at 13:35 | vote | accept | qifeng618 | ||
| Jun 27 at 15:01 | |||||
| Jun 27 at 12:06 | comment | added | qifeng618 | Thank you very much for your answer, Dr. Carlo Beenakker | |
| Jun 27 at 5:52 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
deleted 55 characters in body
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| Jun 27 at 5:45 | history | answered | Carlo Beenakker | CC BY-SA 4.0 |