式
π τ
円の1/4を成す角度
π
2
rad
{\displaystyle {\color {orangered}{\frac {\pi }{2}}}{\text{ rad}}}
τ
4
rad
{\displaystyle {\color {orangered}{\frac {\tau }{4}}}{\text{ rad}}}
円周
C
=
2
π
r
{\displaystyle C={\color {orangered}2\pi }r}
C
=
τ
r
{\displaystyle C={\color {orangered}\tau }r}
[ 注 2]
円の面積
A
=
π
r
2
{\displaystyle A={\color {orangered}\pi }r^{2}}
A
=
1
2
τ
r
2
{\displaystyle A={\color {orangered}{\frac {1}{2}}\tau }r^{2}}
単位円周半径を持つ正n角形 の面積
A
=
n
2
sin
2
π
n
{\displaystyle A={\frac {n}{2}}\sin {\frac {\color {orangered}2\pi }{n}}}
A
=
n
2
sin
τ
n
{\displaystyle A={\frac {n}{2}}\sin {\frac {\color {orangered}\tau }{n}}}
n球とn球の体積再帰関係
V
n
(
r
)
=
r
n
S
n
−
1
(
r
)
{\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}
S
n
(
r
)
=
2
π
r
V
n
−
1
(
r
)
{\displaystyle S_{n}(r)={\color {orangered}2\pi }rV_{n-1}(r)}
V
n
(
r
)
=
r
n
S
n
−
1
(
r
)
{\displaystyle V_{n}(r)={\frac {r}{n}}S_{n-1}(r)}
S
n
(
r
)
=
τ
r
V
n
−
1
(
r
)
{\displaystyle S_{n}(r)={\color {orangered}\tau }rV_{n-1}(r)}
[ 注 3]
コーシーの積分公式
f
(
a
)
=
1
2
π
i
∮
γ
f
(
z
)
z
−
a
d
z
{\displaystyle f(a)={\frac {1}{{\color {orangered}2\pi }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz}
f
(
a
)
=
1
τ
i
∮
γ
f
(
z
)
z
−
a
d
z
{\displaystyle f(a)={\frac {1}{{\color {orangered}\tau }i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz}
標準正規分布 の確率密度関数
φ
(
x
)
=
1
2
π
e
−
x
2
2
{\displaystyle \varphi (x)={\frac {1}{\sqrt {\color {orangered}2\pi }}}e^{-{\frac {x^{2}}{2}}}}
φ
(
x
)
=
1
τ
e
−
x
2
2
{\displaystyle \varphi (x)={\frac {1}{\sqrt {\color {orangered}\tau }}}e^{-{\frac {x^{2}}{2}}}}
スターリングの近似
n
!
∼
2
π
n
(
n
e
)
n
{\displaystyle n!\sim {\sqrt {{\color {orangered}2\pi }n}}\left({\frac {n}{e}}\right)^{n}}
n
!
∼
τ
n
(
n
e
)
n
{\displaystyle n!\sim {\sqrt {{\color {orangered}\tau }n}}\left({\frac {n}{e}}\right)^{n}}
πn乗根
e
2
π
i
k
n
=
cos
2
k
π
n
+
i
sin
2
k
π
n
{\displaystyle e^{{\color {orangered}2\pi }i{\frac {k}{n}}}=\cos {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}+i\sin {\frac {{\color {orangered}2}k{\color {orangered}\pi }}{n}}}
e
τ
i
k
n
=
cos
k
τ
n
+
i
sin
k
τ
n
{\displaystyle e^{{\color {orangered}\tau }i{\frac {k}{n}}}=\cos {\frac {k{\color {orangered}\tau }}{n}}+i\sin {\frac {k{\color {orangered}\tau }}{n}}}
プランク定数
h
=
2
π
ℏ
{\displaystyle h={\color {orangered}2\pi }\hbar }
h
=
τ
ℏ
{\displaystyle h={\color {orangered}\tau }\hbar }
角周波数
ω
=
2
π
f
{\displaystyle \omega ={\color {orangered}2\pi }f}
ω
=
τ
f
{\displaystyle \omega ={\color {orangered}\tau }f}
逆格子ベクトル
G
m
⋅
R
n
=
2
π
N
m
n
{\displaystyle \mathbf {G} _{m}\cdot \mathbf {R} _{n}={\color {orangered}2\pi }N_{mn}}
G
m
⋅
R
n
=
τ
N
m
n
{\displaystyle \mathbf {G} _{m}\cdot \mathbf {R} _{n}={\color {orangered}\tau }N_{mn}}
断面二次極モーメント
I
p
=
π
d
4
32
{\displaystyle I_{p}={\frac {{\color {orangered}\pi }d^{4}}{\color {blue}32}}}
I
p
=
τ
d
4
64
=
τ
(
d
2
)
4
{\displaystyle I_{p}={\frac {{\color {orangered}\tau }d^{4}}{\color {blue}64}}={\color {orangered}\tau }\left({\frac {d}{2}}\right)^{4}}
フーリエ変換 ・フーリエ逆変換
f
^
(
ξ
)
=
∫
−
∞
∞
f
(
x
)
e
−
2
π
i
x
ξ
d
x
{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-{\color {orangered}2\pi }ix\xi }\,dx}
f
(
x
)
=
∫
−
∞
∞
f
^
(
ξ
)
e
2
π
i
x
ξ
d
ξ
{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{{\color {orangered}2\pi }ix\xi }\,d\xi }
f
^
(
ξ
)
=
∫
−
∞
∞
f
(
x
)
e
−
τ
i
x
ξ
d
x
{\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }f(x)e^{-{\color {orangered}\tau }ix\xi }\,dx}
f
(
x
)
=
∫
−
∞
∞
f
^
(
ξ
)
e
τ
i
x
ξ
d
ξ
{\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{{\color {orangered}\tau }ix\xi }\,d\xi }
無限乗積
∏
k
=
1
∞
k
=
2
π
{\displaystyle \prod _{k=1}^{\infty }k={\sqrt {\color {orangered}2\pi }}}
∏
p
=
(
2
π
)
2
{\displaystyle \prod _{p}=({\color {orangered}2\pi })^{2}}
∏
k
=
1
∞
k
=
τ
{\displaystyle \prod _{k=1}^{\infty }k={\sqrt {\color {orangered}\tau }}}
∏
p
=
τ
2
{\displaystyle \prod _{p}={\color {orangered}\tau }^{2}}
