ネイピア数 (Napier's constant; 英語では、むしろ Euler's number と言うが、オイラーの定数 Euler's constant やオイラー数列 と紛らわしい) は、数学定数の一つで、e と書かれる。自然対数 の底である[ 1]
値 [ ] e は無理数であり、超越数である。
十進数: 2.71828182845904523536...
連分数
e
=
2
+
1
1
+
1
2
+
2
3
+
3
4
+
4
5
+
⋱
=
2
+
2
2
+
3
3
+
4
4
+
5
5
+
6
6
+
⋱
{\displaystyle e= 2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}} = 2+\cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}}
e
=
2
+
1
1
+
2
5
+
1
10
+
1
14
+
1
18
+
⋱
=
1
+
2
1
+
1
6
+
1
10
+
1
14
+
1
18
+
⋱
{\displaystyle e = 2+\cfrac{1}{1+\cfrac{2}{5+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}} = 1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}}}}
e
x
/
y
=
1
+
2
x
2
y
−
x
+
x
2
6
y
+
x
2
10
y
+
x
2
14
y
+
x
2
18
y
+
⋱
{\displaystyle e^{x/y} = 1+\cfrac{2x} {2y-x+\cfrac{x^2} {6y+\cfrac{x^2} {10y+\cfrac{x^2} {14y+\cfrac{x^2} {18y+\ddots}}}}}}
極限
lim
n
→
∞
(
1
+
1
n
)
n
=
e
{\displaystyle \lim_{n \to \infty} (1 + {1\over n})^n = e}
lim
n
→
−
∞
(
1
+
1
n
)
n
=
e
{\displaystyle \lim_{n \to -\infty} (1 + {1\over n})^n = e}
lim
n
→
±
∞
(
1
−
1
n
)
−
n
=
e
{\displaystyle \lim_{n \to \pm\infty} (1 - {1\over n})^{-n} = e}
lim
n
→
±
∞
(
1
−
1
n
)
n
=
1
e
{\displaystyle \lim_{n \to \pm\infty} (1 - {1\over n})^n = \frac{1}{e}}
lim
n
→
±
∞
(
1
+
1
n
)
−
n
=
1
e
{\displaystyle \lim_{n \to \pm\infty} (1 + {1\over n})^{-n} = \frac{1}{e}}
lim
n
→
0
(
1
+
n
)
1
/
n
=
e
{\displaystyle \lim_{n \to 0} (1 + n)^{1/n} = e}
lim
n
→
0
(
1
−
n
)
−
1
/
n
=
e
{\displaystyle \lim_{n \to 0} (1 - n)^{-1/n} = e}
lim
n
→
0
(
1
−
n
)
1
/
n
=
1
e
{\displaystyle \lim_{n \to 0} (1 - n)^{1/n} = \frac{1}{e}}
lim
n
→
0
(
1
+
n
)
−
1
/
n
=
1
e
{\displaystyle \lim_{n \to 0} (1 + n)^{-1/n} = \frac{1}{e}}
lim
n
→
∞
(
(
n
+
1
)
n
+
1
n
n
−
n
n
(
n
−
1
)
n
−
1
)
=
e
{\displaystyle \lim_{n \to \infty} (\frac{(n+1)^{n+1}}{n^n}-\frac{n^n}{(n-1)^{n-1}}) = e}
lim
n
→
∞
(
n
!
)
1
n
n
=
1
e
{\displaystyle \lim_{n \to \infty} \frac{(n!)^{1\over n}}{n} = {1\over e}}
e
=
lim
n
→
∞
(
p
n
#
)
1
/
p
n
{\displaystyle e= \lim_{n \to \infty}(p_n \#)^{1/p_n} }
素数 、#は素数階乗 )
e
=
lim
n
→
∞
n
π
(
n
)
/
n
{\displaystyle e= \lim_{n \to \infty}n^{\pi(n)/n} }
テイラー展開:
e
=
∑
n
=
0
∞
1
n
!
{\displaystyle e = \sum_{n = 0}^{\infty} \frac{1}{n!}}
e
x
=
∑
n
=
0
∞
x
n
n
!
{\displaystyle e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!} }
総和
1
e
=
∑
n
=
0
∞
(
−
1
)
n
n
!
{\displaystyle \frac{1}{e} = \sum_{n=0}^\infty \frac{(-1)^n}{n!}}
e
=
∑
n
=
0
∞
2
n
+
1
(
2
n
)
!
{\displaystyle e = \sum_{n=0}^\infty \frac{2n+1}{(2n)!}}
e
=
1
2
∑
n
=
0
∞
n
+
1
n
!
{\displaystyle e = \frac{1}{2}\sum_{n=0}^\infty \frac{n+1}{n!}}
e
=
2
∑
n
=
0
∞
n
+
1
(
2
n
+
1
)
!
{\displaystyle e = 2\sum_{n=0}^\infty \frac{n+1}{(2n+1)!}}
e
=
∑
n
=
0
∞
3
−
4
n
2
(
2
n
+
1
)
!
{\displaystyle e = \sum_{n=0}^\infty \frac{3-4n^2}{(2n+1)!}}
e
=
∑
n
=
0
∞
(
3
n
)
2
+
1
(
3
n
)
!
{\displaystyle e = \sum_{n=0}^\infty \frac{(3n)^2+1}{(3n)!}}
e
=
∑
n
=
0
∞
4
n
+
3
2
2
n
+
1
(
2
n
+
1
)
!
{\displaystyle \sqrt e = \sum_{n=0}^\infty \frac{4n+3}{2^{2n+1}(2n+1)!}}
e
=
[
−
12
π
2
∑
k
=
1
∞
1
k
2
cos
(
9
k
π
+
k
2
π
2
−
9
)
]
−
1
/
3
{\displaystyle e = \left [ -\frac{12}{\pi^2} \sum_{k=1}^\infty \frac{1}{k^2} \ \cos \left ( \frac{9}{k\pi+\sqrt{k^2\pi^2-9}} \right ) \right ]^{-1/3} }
e
=
∑
k
=
1
∞
k
n
B
n
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^n}{B_n(k!)}}
ベル数 )
e
=
∑
k
=
1
∞
k
k
!
=
∑
k
=
1
∞
1
(
k
−
1
)
!
=
∑
k
=
0
∞
1
k
!
{\displaystyle e = \sum_{k=1}^\infty \frac{k}{k!} = \sum_{k=1}^\infty \frac{1}{(k-1)!} = \sum_{k=0}^\infty \frac{1}{k!}}
e
=
∑
k
=
1
∞
k
2
2
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^2}{2(k!)}}
e
=
∑
k
=
1
∞
k
3
5
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^3}{5(k!)}}
e
=
∑
k
=
1
∞
k
4
15
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^4}{15(k!)}}
e
=
∑
k
=
1
∞
k
5
52
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^5}{52(k!)}}
e
=
∑
k
=
1
∞
k
6
203
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^6}{203(k!)}}
e
=
∑
k
=
1
∞
k
7
877
(
k
!
)
{\displaystyle e = \sum_{k=1}^\infty \frac{k^7}{877(k!)}}
総乗
2
∏
n
=
1
∞
∏
m
=
2
n
2
n
+
1
−
1
(
2
m
(
2
m
+
2
)
(
2
m
+
1
)
2
)
1
2
n
+
1
=
e
2
{\displaystyle \sqrt2 \prod_{n=1}^\infty \prod_{m=2^n}^{2^{n+1}-1} (\frac{2m(2m+2)}{(2m+1)^2})^{\frac{1}{2^{n+1}}}= \frac{e}{2}}
e
=
2
⋅
2
(
ln
(
2
)
−
1
)
2
⋯
2
ln
(
2
)
−
1
⋅
2
(
ln
(
2
)
−
1
)
3
⋯
{\displaystyle e = \frac{2\cdot 2^{(\ln(2)-1)^2} \cdots}{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^3}\cdots }}
双曲線関数
e
x
=
sinh
(
x
)
+
cosh
(
x
)
{\displaystyle e^x = \sinh(x) + \cosh(x)}
性質 [ ] e =
a
b
{\displaystyle \frac{a}{b}}
b
!
⋅
e
=
(
b
!
+
b
!
1
!
+
b
!
2
!
+
b
!
3
!
+
⋯
+
b
!
b
!
)
+
{
b
!
(
b
+
1
)
!
+
b
!
(
b
+
2
)
!
+
b
!
(
b
+
3
)
!
+
⋯
}
{\displaystyle b! \cdot e = \left(b! + \frac{b!}{1!} + \frac{b!}{2!} + \frac{b!}{3!} + \cdots + \frac{b!}{b!}\right)+ \left\{ \frac{b!}{(b+1)!} + \frac{b!}{(b+2)!} + \frac{b!}{(b+3)!} + \cdots \right\}}
左辺は
b
!
⋅
e
=
b
!
⋅
a
b
=
a
(
b
−
1
)
!
{\displaystyle b! \cdot e = b! \cdot \frac{a}{b} = a(b-1)!}
b
!
b
!
{\displaystyle \frac{b!}{b!}}
b
!
(
b
+
1
)
!
{\displaystyle \frac{b!}{(b+1)!}}
{
b
!
(
b
+
1
)
!
+
b
!
(
b
+
2
)
!
+
b
!
(
b
+
3
)
!
+
⋯
}
{\displaystyle \left\{ \frac{b!}{(b+1)!} + \frac{b!}{(b+2)!} + \frac{b!}{(b+3)!} + \cdots \right\} }
=
1
(
b
+
1
)
+
1
(
b
+
1
)
(
b
+
2
)
+
1
(
b
+
1
)
(
b
+
2
)
(
b
+
3
)
+
⋯
<
1
2
+
1
2
2
+
1
2
3
+
⋯
=
1
{\displaystyle = \frac{1}{(b+1)} + \frac{1}{(b+1)(b+2)} + \frac{1}{(b+1)(b+2)(b+3)} + \cdots < \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots = 1}
と 1 未満になる。したがって ( ) 内と { } 内を足した右辺は自然数でないことになり、左辺が自然数という結果と矛盾する。
ゆえに
e
=
a
b
{\displaystyle e = \frac{a}{b}}
超越性 [ ] 暗記 [ ] Examples of mnemonics (Gardner 1959, 1991) include:
"By omnibus I traveled to Brooklyn" (6 digits).
"To disrupt a playroom is commonly a practice of children" (10 digits).
"It enables a numskull to memorize a quantity of numerals" (10 digits).
"I'm forming a mnemonic to remember a function in analysis" (10 digits).
"He repeats: I shouldn't be tippling, I shouldn't be toppling here!" (11 digits).
"In showing a painting to probably a critical or venomous lady, anger dominates. O take guard, or she raves and shouts" (21 digits). Here, the word "O" stands for the number 0.
A much more extensive mnemonic giving 40 digits is
"We present a mnemonic to memorize a constant so exciting that Euler exclaimed: '!' when first it was found, yes, loudly '!'. My students perhaps will compute , use power or Taylor series, an easy summation formula, obvious, clear, elegant!"
関係項目 [ ] 出典 [ ] 
![{\displaystyle e=\left[-{\frac {12}{\pi ^{2}}}\sum _{k=1}^{\infty }{\frac {1}{k^{2}}}\ \cos \left({\frac {9}{k\pi +{\sqrt {k^{2}\pi ^{2}-9}}}}\right)\right]^{-1/3}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/5128e41fb2adbf92bd0c875101b2da9d90f9e7d9)
