[Math]Pi(1)
数学知识忘地太快，在博客记录一下pi的生成。
- 100 Decimal places
- 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
- Approximations
- 22/7 3 decimal places (used by Egyptians around 1000BC)
- 666/212 4 decimal places
- 355/113 6 decimal places
- 104348/33215 8 decimal places
- Series Expansions
- English mathematician John Wallis in 1655.
4 * 4 * 6 * 6 * 8 * 8 * 10 * 10 * 12 * 12 .....
pi = 8 * -------------------------------------------------
3 * 3 * 5 * 5 * 7 * 7 * 9 * 9 * 11 * 11 ....
- Scottish mathematician and astronomer James Gregory in 1671
pi = 4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ....)
- Swiss mathematician Leonard Euler.
pi = sqrt(12 - (12/22) + (12/32) - (12/42) + (12/52) .... ) …… (1)
pi = sqrt[6 * ( 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + ...)] …… (2)
下面则试证一下 Gregory’s Series
1. Taylor series
egin{equation}label{E1}
fleft( x
ight) = sumlimits_{n = 0}^infty {frac{{ { f^{left( n
ight)}}left( a
ight)}}{{n!}}} {left( {x - a}
ight)^n}
end{equation}
2. Maclaurin series
egin{equation}label{E2}
fleft( x
ight) = sumlimits_{n = 0}^infty {frac{ f^{left( n
ight)}left( 0
ight) }{n!} } { x^n }
end{equation}
3. arctan(x)一阶导数
egin{align*}
&y = f left( x
ight) = arctan left( x
ight) \
&x = tan left( y
ight)
end{align*}
egin{align*}
Longrightarrow dx &= sec^{2}y * dy \
f^{ prime }{ left( x
ight) }&= { frac {dx}{dy} } = {frac{1}{ x^{2}+1 } }
end{align*}
4. 推导过程
(1).y=arctan(x)的n阶导可以用下面的方法求得：
egin{align*}
ecause &arctan left( x
ight) = int
olimits_0^x frac{1}{ 1+t^{2} } \,dt \
&frac{1}{1+x^{2} } = frac{1}{2}( frac{1}{1-ix} + frac{1}{1+ix} ) \
herefore &arctan left( x
ight) = frac{1}{2}i left[ ln (1-ix) -ln (1+ix)
ight]
end{align*}
(2).若按原始方法，得先记住分数函数的求导方式：
$$ left( frac { f left( x ight) } { g left( x ight)} ight)^{prime} = frac { { f^{ prime } left( x ight) } { g left( x ight) } - { f left( x ight) } { g^{ prime } left( x ight) } } { g^{2} left( x ight) } $$
(3).f(x)的n阶导数
egin{align*}
& f ^{left( 1
ight)}left( x
ight) = {frac{1}{ x^{2}+1 } } \
& f ^{left( 2
ight)}left( x
ight) = {frac{-2x}{ left(x^{2}+1
ight)^{2} } } \
& f ^{left( 3
ight)}left( x
ight) = {frac{2left( 3x^{2}-1
ight) }{ left(x^{2}+1
ight)^{3} } } \
& f ^{left( 4
ight)}left( x
ight) = {frac{-24xleft(x^{2}-1
ight) }{ left(x^{2}+1
ight)^{4} } } \
& f ^{left( 5
ight)}left( x
ight) = {frac{24left(5x^{4}-10x^{2}+1
ight) }{ left(x^{2}+1
ight)^{5} } } \
& ...\
& f ^{left( n
ight)}left( x
ight) = frac {1}{2} (-1)^{n} i left[ (-i+x)^{-n}-(i+x)^{-n}
ight] (n-1)! \
& ...\
end{align*}
(4).f(x) Taylor Series Expansion 的系数
egin{align*}
k_{1} &= frac{ f ^{left( 1
ight)}left( 0
ight) } { 1! } = 1\
k_{2} &= frac{ f ^{left( 2
ight)}left( 0
ight) } { 2! } = 0\
k_{3} &= frac{ f ^{left( 3
ight)}left( 0
ight) } { 3! } = frac {-1}{3}\
k_{4} &= frac{ f ^{left( 4
ight)}left( 0
ight) } { 4! } = 0\
k_{5} &= frac{ f ^{left( 5
ight)}left( 0
ight) } { 5! } = frac {1}{5}\
& ...\
end{align*}
5. get the conclusion, Maclaurin Series.
『Gregory's series』 or 『Leibniz's series』
egin{align*}
ecause arctan left( x
ight) &= sum limits_{n=0}^{infty} (-1)^{n} { frac{1}{2n+1} } x^{2n+1} \
&= x - frac{1}{3}x^{3} + frac{1}{5}x^{5} - frac{1}{7}x^{7} + ...\
herefore arctan left( 1
ight) &= 1-frac{1}{3} + frac{1}{5} - frac{1}{7} + frac{1}{9} -frac{1}{11}+... =frac{ pi }{4}
end{align*}