f
(
x
)
{\displaystyle f(x)}
dom
f
{\displaystyle \operatorname {dom} f}
f
⋆
(
x
⋆
)
{\displaystyle f^{\star }(x^{\star })}
dom
f
⋆
{\displaystyle \operatorname {dom} f^{\star }}
조건
a
f
(
x
)
{\displaystyle af(x)}
dom
f
{\displaystyle \operatorname {dom} f}
a
f
⋆
(
x
⋆
/
a
)
{\displaystyle af^{\star }(x^{\star }/a)}
a
⋅
dom
f
⋆
{\displaystyle a\cdot \operatorname {dom} f^{\star }}
a
>
0
{\displaystyle a>0}
f
(
a
x
)
{\displaystyle f(ax)}
a
−
1
⋅
dom
f
{\displaystyle a^{-1}\cdot \operatorname {dom} f}
f
⋆
(
x
⋆
/
a
)
{\displaystyle f^{\star }(x^{\star }/a)}
a
⋅
dom
f
⋆
{\displaystyle a\cdot \operatorname {dom} f^{\star }}
a
>
0
{\displaystyle a>0}
f
(
x
)
+
a
{\displaystyle f(x)+a}
dom
f
{\displaystyle \operatorname {dom} f}
f
⋆
(
x
⋆
)
−
a
{\displaystyle f^{\star }(x^{\star })-a}
dom
f
⋆
{\displaystyle \operatorname {dom} f^{\star }}
a
∈
R
{\displaystyle a\in \mathbb {R} }
f
(
x
−
a
)
{\displaystyle f(x-a)}
a
+
dom
f
{\displaystyle a+\operatorname {dom} f}
f
⋆
(
x
⋆
)
+
a
x
⋆
{\displaystyle f^{\star }(x^{\star })+ax^{\star }}
dom
f
⋆
{\displaystyle \operatorname {dom} f^{\star }}
a
∈
R
{\displaystyle a\in \mathbb {R} }
f
(
x
)
+
a
x
{\displaystyle f(x)+ax}
dom
f
{\displaystyle \operatorname {dom} f}
f
⋆
(
x
⋆
−
a
)
{\displaystyle f^{\star }(x^{\star }-a)}
a
+
dom
f
⋆
{\displaystyle a+\operatorname {dom} f^{\star }}
a
∈
R
{\displaystyle a\in \mathbb {R} }
f
(
x
)
+
g
(
x
)
{\displaystyle f(x)+g(x)}
dom
f
∩
dom
g
{\displaystyle \operatorname {dom} f\cap \operatorname {dom} g}
(
f
⋆
⋆
inf
g
⋆
)
(
x
⋆
)
{\displaystyle (f^{\star }\star _{\text{inf}}g^{\star })(x^{\star })}
dom
f
⋆
+
dom
g
⋆
{\displaystyle \operatorname {dom} f^{\star }+\operatorname {dom} g^{\star }}
(
f
⋆
inf
g
)
(
x
)
=
inf
y
{
f
(
x
−
y
)
+
g
(
y
)
}
{\displaystyle (f\star _{\text{inf}}g)(x)=\inf _{y}\{f(x-y)+g(y)\}}
(
f
⋆
inf
g
)
(
x
)
{\displaystyle (f\star _{\text{inf}}g)(x)}
dom
f
+
dom
g
{\displaystyle \operatorname {dom} f+\operatorname {dom} g}
f
⋆
(
x
⋆
)
+
g
⋆
(
x
⋆
)
{\displaystyle f^{\star }(x^{\star })+g^{\star }(x^{\star })}
dom
f
⋆
∩
dom
g
⋆
{\displaystyle \operatorname {dom} f^{\star }\cap \operatorname {dom} g^{\star }}
(
f
⋆
inf
g
)
(
x
)
=
inf
y
{
f
(
x
−
y
)
+
g
(
y
)
}
{\displaystyle (f\star _{\text{inf}}g)(x)=\inf _{y}\{f(x-y)+g(y)\}}
a
x
+
b
{\displaystyle ax+b}
R
{\displaystyle \mathbb {R} }
−
b
{\displaystyle -b}
{
a
}
{\displaystyle \{a\}}
|
x
|
p
/
p
{\displaystyle |x|^{p}/p}
R
{\displaystyle \mathbb {R} }
|
x
⋆
|
p
⋆
/
p
⋆
{\displaystyle |x^{\star }|^{p^{\star }}/p^{\star }}
R
{\displaystyle \mathbb {R} }
1
/
p
+
1
/
p
⋆
=
1
{\displaystyle 1/p+1/p^{\star }=1}
p
>
1
{\displaystyle p>1}
−
x
p
/
p
{\displaystyle -x^{p}/p}
[
0
,
∞
)
{\displaystyle [0,\infty )}
−
|
x
⋆
|
p
⋆
/
p
⋆
{\displaystyle -|x^{\star }|^{p^{\star }}/p^{\star }}
(
−
∞
,
0
]
{\displaystyle (-\infty ,0]}
1
/
p
+
1
/
p
⋆
=
1
{\displaystyle 1/p+1/p^{\star }=1}
p
<
1
{\displaystyle p<1}
exp
(
x
)
{\displaystyle \exp(x)}
R
{\displaystyle \mathbb {R} }
x
⋆
(
ln
(
x
⋆
)
−
1
)
{\displaystyle x^{\star }(\ln(x^{\star })-1)}
R
+
{\displaystyle \mathbb {R} ^{+}}
x
ln
(
x
)
{\displaystyle x\ln(x)}
R
+
{\displaystyle \mathbb {R} ^{+}}
exp
(
x
−
1
)
{\displaystyle \exp(x-1)}
R
{\displaystyle \mathbb {R} }
−
1
/
2
−
ln
x
{\displaystyle -1/2-\ln x}
R
+
{\displaystyle \mathbb {R} ^{+}}
−
1
/
2
−
ln
|
x
⋆
|
{\displaystyle -1/2-\ln |x^{\star }|}
R
−
{\displaystyle \mathbb {R} ^{-}}
x
exp
(
x
+
1
)
{\displaystyle x\exp(x+1)}
R
{\displaystyle \mathbb {R} }
x
⋆
(
W
(
x
⋆
)
−
1
)
2
/
W
(
x
⋆
)
{\displaystyle x^{\star }(W(x^{\star })-1)^{2}/W(x^{\star })}
[
−
1
/
e
,
∞
)
{\displaystyle [-1/e,\infty )}
W
{\displaystyle W}
람베르트 W 함수
![{\displaystyle (-\infty ,0]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0241c015ef4c611c9c9aeafb395e7c4a16178405)
