劉維爾定理 揭示了具有初等 原函數 的初等函數的本質特徵。其最早由約瑟夫·劉維爾 於十九世紀三四十年代提出，經後人推廣到一般的微分域上[ 1] 常微分方程組 初等首次積分 的研究上。[ 2] [ 3]
初等函數的原函數並不總是初等函數，例如
e
−
x
2
{\displaystyle e^{-x^{2}}}
誤差函數 ，無法用初等函數表達出來。 其它常見的例子還有
sin
(
x
)
/
x
{\displaystyle \sin(x)/x}
x
x
{\displaystyle x^{x}}
1
/
ln
(
x
)
{\displaystyle 1/\ln(x)}
劉維爾定理指出，一個初等函數如果有初等的原函數，那麼一定能寫成同一個微分域 的函數加上有限項該域上函數的對數的線性組合，否則即表明不存在初等的原函數。
一個域
F
{\displaystyle F}
δ
{\displaystyle \delta }
(
F
,
δ
)
{\displaystyle (F,\delta )}
微分域 。若對於
∀
f
,
g
∈
F
{\displaystyle \forall f,g\in F}
δ
(
f
+
g
)
=
δ
(
f
)
+
δ
(
g
)
,
δ
(
f
g
)
=
δ
(
f
)
g
+
f
δ
(
g
)
{\displaystyle \delta (f+g)=\delta (f)+\delta (g),\quad \delta (fg)=\delta (f)g+f\delta (g)}
由上式可以得到通常導數的一些性質。
δ
(
g
n
)
=
n
g
n
−
1
δ
(
g
)
{\displaystyle \delta (g^{n})=ng^{n-1}\delta (g)}
δ
(
f
g
)
=
δ
(
f
)
g
+
f
δ
(
1
g
)
=
δ
(
f
)
g
−
f
g
2
δ
(
g
)
{\displaystyle \delta ({\frac {f}{g}})={\frac {\delta (f)}{g}}+f\delta ({\frac {1}{g}})={\frac {\delta (f)}{g}}-{\frac {f}{g^{2}}}\delta (g)}
(
F
,
δ
)
{\displaystyle (F,\delta )}
C
o
n
(
F
,
δ
)
=
{
f
∈
F
|
δ
f
=
0
}
{\displaystyle \mathrm {Con} (F,\delta )=\{f\in F|\delta f=0\}}
為該微分域的常數域。
h
∈
K
{\displaystyle h\in K}
K 是 F 的微分域擴張
K
=
F
(
h
)
{\displaystyle K=F(h)}
h
{\displaystyle h}
F
{\displaystyle F}
h
{\displaystyle h}
F
{\displaystyle F}
F
{\displaystyle F}
p
(
t
)
(
∈
F
[
t
]
)
{\displaystyle p(t)(\in F[t])}
p
(
h
)
=
0
{\displaystyle p(h)=0}
p
(
t
)
{\displaystyle p(t)}
p
(
t
)
=
0
{\displaystyle p(t)=0}
t
(
x
)
=
h
(
x
)
{\displaystyle t(x)=h(x)}
解析分支 ）。稱這種情況為代數擴張。
h
{\displaystyle h}
F
{\displaystyle F}
超越 元素，且
δ
h
∈
F
{\displaystyle \delta h\in F}
f
∈
F
{\displaystyle f\in F}
δ
(
ln
(
f
)
)
=
δ
(
f
)
/
f
∈
F
{\displaystyle \delta (\ln(f))=\delta (f)/f\in F}
h
{\displaystyle h}
F
{\displaystyle F}
h
{\displaystyle h}
F
{\displaystyle F}
δ
h
/
h
∈
F
{\displaystyle \delta h/h\in F}
f
∈
F
{\displaystyle f\in F}
δ
(
exp
(
f
)
)
/
exp
(
f
)
=
δ
(
f
)
∈
F
{\displaystyle \delta (\exp(f))/\exp(f)=\delta (f)\in F}
h
{\displaystyle h}
F
{\displaystyle F}
初等擴張 是指接連進行如上的擴張得到的微分域
F
(
h
1
,
.
.
.
,
h
n
)
{\displaystyle F(h_{1},...,h_{n})}
h
j
{\displaystyle h_{j}}
F
(
h
1
,
.
.
.
,
h
j
−
1
)
{\displaystyle F(h_{1},...,h_{j-1})}
f
(
x
)
{\displaystyle f(x)}
初等函數 若它在微分域
(
C
(
x
)
,
d
/
d
x
)
{\displaystyle (\mathbb {C} (x),\mathrm {d} /\mathrm {d} x)}
以下為劉維爾第一定理（Theorem of Liouville-first statement）。
設
F
{\displaystyle F}
K
{\displaystyle K}
F
{\displaystyle F}
C
o
n
(
K
,
δ
)
=
C
o
n
(
F
,
δ
)
{\displaystyle \mathrm {Con} (K,\delta )=\mathrm {Con} (F,\delta )}
f
∈
F
{\displaystyle f\in F}
g
∈
K
{\displaystyle g\in K}
δ
g
=
f
{\displaystyle \delta g=f}
[ 4] [ 5]
g
=
c
1
ln
(
u
1
)
+
⋯
+
c
n
ln
(
u
n
)
+
v
.
{\displaystyle g=c_{1}\ln(u_{1})+\dotsb +c_{n}\ln(u_{n})+v.}
其中
c
1
,
.
.
.
,
c
n
∈
C
o
n
(
F
,
δ
)
{\displaystyle c_{1},...,c_{n}\in \mathrm {Con} (F,\delta )}
u
1
,
.
.
.
,
u
n
,
v
∈
F
{\displaystyle u_{1},...,u_{n},v\in F}
以下為劉維爾第二定理（Theorem of Liouville-second statement），又稱強劉維爾定理（Strong Liouville theorem）。
設
F
{\displaystyle F}
B
=
C
o
n
(
F
,
δ
)
{\displaystyle B=\mathrm {Con} (F,\delta )}
g
{\displaystyle g}
F
{\displaystyle F}
δ
g
=
f
∈
F
{\displaystyle \delta g=f\in F}
f
=
c
1
δ
u
1
u
1
+
⋯
+
c
n
δ
u
n
u
n
+
δ
v
.
{\displaystyle f=c_{1}{\frac {\delta u_{1}}{u_{1}}}+\dotsb +c_{n}{\frac {\delta u_{n}}{u_{n}}}+\delta v.}
其中
c
1
,
.
.
.
,
c
n
∈
B
¯
{\displaystyle c_{1},...,c_{n}\in {\bar {B}}}
v
∈
F
{\displaystyle v\in F}
u
1
,
.
.
.
,
u
n
,
v
∈
B
¯
F
{\displaystyle u_{1},...,u_{n},v\in {\bar {B}}F}
B
¯
{\displaystyle {\bar {B}}}
B
{\displaystyle B}
代數閉域 .
每個
F
{\displaystyle F}
B
¯
F
{\displaystyle {\bar {B}}F}
自同構 交換求和的順序。
註：對於通常所說的初等函數 ，
C
o
n
(
K
,
δ
)
=
C
o
n
(
F
,
δ
)
=
C
{\displaystyle \mathrm {Con} (K,\delta )=\mathrm {Con} (F,\delta )=\mathbb {C} }
C
o
n
(
F
,
δ
)
=
R
{\displaystyle \mathrm {Con} (F,\delta )=\mathbb {R} }
1
/
(
x
2
+
1
)
{\displaystyle 1/(x^{2}+1)}
虛數 。
例如複數域 上的有理函數 域
C
(
x
)
{\displaystyle \mathbb {C} (x)}
導數 即構成了一個微分域
(
C
(
x
)
,
d
/
d
x
)
{\displaystyle (\mathbb {C} (x),\mathrm {d} /\mathrm {d} x)}
C
{\displaystyle \mathbb {C} }
函數
1
/
x
∈
C
(
x
)
{\displaystyle 1/x\in \mathbb {C} (x)}
ln
(
x
)
+
C
{\displaystyle \ln(x)+C}
(
C
(
x
)
,
d
/
d
x
)
{\displaystyle (\mathbb {C} (x),\mathrm {d} /\mathrm {d} x)}
x
,
C
∈
C
(
x
)
,
1
∈
C
{\displaystyle x,C\in \mathbb {C} (x),1\in \mathbb {C} }
1
/
(
x
2
+
1
)
∈
C
(
x
)
{\displaystyle 1/(x^{2}+1)\in \mathbb {C} (x)}
反正切函數 可以表達成對數的形式
arctan
(
x
)
+
C
=
−
i
2
ln
1
+
i
x
1
−
i
x
+
C
{\displaystyle \arctan(x)+C=-{\frac {i}{2}}\ln {\frac {1+ix}{1-ix}}+C}
顯然也有
C
,
1
+
i
x
1
−
i
x
∈
C
(
x
)
,
−
i
2
∈
C
{\displaystyle C,{\frac {1+ix}{1-ix}}\in \mathbb {C} (x),-{\frac {i}{2}}\in \mathbb {C} }
下面考慮
f
(
x
)
=
1
/
(
x
ln
(
x
)
)
{\displaystyle f(x)=1/(x\ln(x))}
C
(
x
)
{\displaystyle \mathbb {C} (x)}
ln
(
x
)
{\displaystyle \ln(x)}
C
(
x
)
{\displaystyle \mathbb {C} (x)}
超越函數 ）。把
ln
(
x
)
{\displaystyle \ln(x)}
C
(
x
)
{\displaystyle \mathbb {C} (x)}
(
F
,
d
/
d
x
)
,
F
=
C
(
x
)
(
ln
(
x
)
)
{\displaystyle (F,\mathrm {d} /\mathrm {d} x),F=\mathbb {C} (x)(\ln(x))}
f
∈
F
{\displaystyle f\in F}
f
(
x
)
{\displaystyle f(x)}
ln
(
ln
(
x
)
)
{\displaystyle \ln(\ln(x))}
f
(
x
)
{\displaystyle f(x)}
F
{\displaystyle F}
f
(
x
)
{\displaystyle f(x)}
事實上，Risch 1969 年的論文表明，對於任意複雜的初等函數，總可以找到適當的包含
f
(
x
)
{\displaystyle f(x)}
F
{\displaystyle F}
C
(
x
)
{\displaystyle \mathbb {C} (x)}
C
(
x
,
x
1
,
.
.
.
,
x
n
)
=
F
{\displaystyle \mathbb {C} (x,x_{1},...,x_{n})=F}
Risch算法 ）。[ 5]
設想我們想知道形如
f
(
x
)
e
g
(
x
)
,
f
(
x
)
,
g
(
x
)
∈
C
(
x
)
{\displaystyle f(x)e^{g(x)},f(x),g(x)\in \mathbb {C} (x)}
a
(
x
)
∈
C
(
x
)
{\displaystyle a(x)\in \mathbb {C} (x)}
f
(
x
)
=
a
′
(
x
)
+
a
(
x
)
g
′
(
x
)
.
{\displaystyle f(x)=a'(x)+a(x)g'(x).}
若存在這樣的
a
(
x
)
{\displaystyle a(x)}
a
(
x
)
e
g
(
x
)
{\displaystyle a(x)e^{g(x)}}
例如對於
e
x
2
{\displaystyle e^{x^{2}}}
f
(
x
)
=
1
,
g
(
x
)
=
x
2
{\displaystyle f(x)=1,g(x)=x^{2}}
1
=
a
′
(
x
)
+
2
x
⋅
a
(
x
)
.
{\displaystyle 1=a'(x)+2x\cdot a(x).}
如果存在這樣的
a
(
x
)
{\displaystyle a(x)}
部分分式展開 ：
a
(
x
)
=
p
(
x
)
+
∑
j
=
1
q
∑
k
=
1
e
j
A
j
k
(
x
−
r
j
)
k
{\displaystyle a(x)=p(x)+\sum _{j=1}^{q}\sum _{k=1}^{e_{j}}{\frac {A_{jk}}{(x-r_{j})^{k}}}}
其中
p
(
x
)
∈
C
[
x
]
{\displaystyle p(x)\in \mathbb {C} [x]}
C
{\displaystyle \mathbb {C} }
r
j
∈
C
{\displaystyle r_{j}\in \mathbb {C} }
a
(
x
)
{\displaystyle a(x)}
A
j
k
∈
C
{\displaystyle A_{jk}\in \mathbb {C} }
a
(
x
)
{\displaystyle a(x)}
2
x
⋅
a
(
x
)
{\displaystyle 2x\cdot a(x)}
p
(
x
)
=
0
{\displaystyle p(x)=0}
1
/
(
x
−
r
j
)
k
{\displaystyle 1/(x-r_{j})^{k}}
e
x
2
{\displaystyle e^{x^{2}}}
藉助完全類似的方法，我們可以證明
e
x
/
x
{\displaystyle e^{x}/x}
1
/
x
=
a
′
+
a
{\displaystyle 1/x=a'+a}
sin
(
x
)
/
x
{\displaystyle \sin(x)/x}
e
x
/
x
{\displaystyle e^{x}/x}
e
e
x
{\displaystyle e^{e^{x}}}
1
/
ln
(
x
)
{\displaystyle 1/\ln(x)}
[ 4]
^ Lützen, J. (1990). Integration in Finite Terms. In Joseph Liouville 1809–1882 (pp. 351-422). Springer New York.
^ Prelle, M. J., & Singer, M. F. (1983). Elementary first integrals of differential equations. Transactions of the American Mathematical Society, 279(1), 215-229.
^ Singer, Michael F. "Liouvillian first integrals of differential equations." Transactions of the American Mathematical Society 333.2 (1992): 673-688.
^ 4.0 4.1 Rosenlicht, M. (1972). Integration in finite terms. American Mathematical Monthly, 963-972.
^ 5.0 5.1 Risch, Robert H. "The problem of integration in finite terms." Transactions of the American Mathematical Society (1969): 167-189.
![{\displaystyle p(t)(\in F[t])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a504368e703f91c41bac3ba24504bd2a2b8eab4)
![{\displaystyle p(x)\in \mathbb {C} [x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8eb6c52fee01c3912e56773472c57dc153a5bdc)
