# 齊次函數

## 正式定義

${\displaystyle f(\alpha \mathbf {v} )=\alpha ^{k}f(\mathbf {v} )}$

## 例子

• 線性函數${\displaystyle f:V\rightarrow W}$ 是一次齊次函數，因為根據線性的定義，對於所有的${\displaystyle \alpha \in F}$ ${\displaystyle \mathbf {v} \in V}$ ，都有：${\displaystyle f(\alpha \mathbf {v} )=\alpha f(\mathbf {v} )}$
• 多線性函數${\displaystyle f:V_{1}\times \ldots \times V_{n}\rightarrow W}$ 是n次齊次函數，因為根據多線性的定義，對於所有的${\displaystyle \alpha \in F}$ ${\displaystyle \mathbf {v} _{1}\in V_{1},\ldots ,\mathbf {v} _{n}\in V_{n}}$ 都有：${\displaystyle f(\alpha \mathbf {v} _{1},\ldots ,\alpha \mathbf {v} _{n})=\alpha ^{n}f(\mathbf {v} _{1},\ldots ,\mathbf {v} _{n})}$
• 從上一個例子中可以看出，兩個巴拿赫空間${\displaystyle X}$ ${\displaystyle Y}$ 之間的函數${\displaystyle f:X\rightarrow Y}$ ${\displaystyle n}$ 弗雷歇導數${\displaystyle n}$ 次齊次函數。
• ${\displaystyle n}$ 單項式定義了齊次函數${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$

${\displaystyle f(x,y,z)=x^{5}y^{2}z^{3}}$

${\displaystyle (\alpha x)^{5}(\alpha y)^{2}(\alpha z)^{3}=\alpha ^{10}x^{5}y^{2}z^{3}}$
• 齊次多項式是由同次數的單項式相加所組成的多項式。例如：
${\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}}$

## 基本定理

• 歐拉定理：假設函數${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ 可導的，且是${\displaystyle k}$ 次齊次函數。那麼：
${\displaystyle \mathbf {x} \cdot \nabla f(\mathbf {x} )=kf(\mathbf {x} )\qquad }$

${\displaystyle f(\alpha \mathbf {x} )=\alpha ^{k}f(\mathbf {x} )}$

${\displaystyle {\frac {\partial }{\partial \alpha x_{1}}}f(\alpha \mathbf {x} ){\frac {\mathrm {d} }{\mathrm {d} \alpha }}(\alpha x_{1})+\cdots +{\frac {\partial }{\partial \alpha {x_{n}}}}f(\alpha \mathbf {x} ){\frac {\mathrm {d} }{\mathrm {d} \alpha }}(\alpha x_{n})=k\alpha ^{k-1}f(\mathbf {x} )}$

${\displaystyle x_{1}{\frac {\partial }{\partial \alpha x_{1}}}f(\alpha \mathbf {x} )+\cdots +x_{n}{\frac {\partial }{\partial \alpha x_{n}}}f(\alpha \mathbf {x} )=k\alpha ^{k-1}f(\mathbf {x} )}$

${\displaystyle \mathbf {x} \cdot \nabla f(\alpha \mathbf {x} )=k\alpha ^{k}f(\mathbf {x} ),\qquad \nabla =({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}})}$

${\displaystyle \alpha =1}$ ，定理即得證。

• 假設${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$ 是可導的，且是${\displaystyle k}$ 階齊次函數。則它的一階偏導數${\displaystyle \partial f/\partial x_{i}}$ ${\displaystyle k-1}$ 階齊次函數。

${\displaystyle f(\alpha \mathbf {x} )=\alpha ^{k}f(\mathbf {x} )}$

${\displaystyle {\frac {\partial }{\partial \alpha x_{i}}}f(\alpha \mathbf {x} ){\frac {\mathrm {d} }{\mathrm {d} x_{i}}}(\alpha x_{i})=\alpha ^{k}{\frac {\partial }{\partial x_{i}}}f(\mathbf {x} ){\frac {\mathrm {d} }{\mathrm {d} x_{i}}}(x_{i})}$

${\displaystyle \alpha {\frac {\partial }{\partial \alpha x_{i}}}f(\alpha \mathbf {x} )=\alpha ^{k}{\frac {\partial }{\partial x_{i}}}f(\mathbf {x} )}$

${\displaystyle {\frac {\partial }{\partial \alpha x_{i}}}f(\alpha \mathbf {x} )=\alpha ^{k-1}{\frac {\partial }{\partial x_{i}}}f(\mathbf {x} )}$ .

## 用於解微分方程

${\displaystyle I(x,y){\frac {\mathrm {d} y}{\mathrm {d} x}}+J(x,y)=0,}$

${\displaystyle x{\frac {\mathrm {d} v}{\mathrm {d} x}}=-{\frac {J(1,v)}{I(1,v)}}-v}$

## 參考文獻

• Blatter, Christian. 20. Mehrdimensionale Differentialrechnung, Aufgaben, 1.. Analysis II (2nd ed.). Springer Verlag. 1979: p. 188. ISBN 3-540-09484-9 （德語）.