这是一个不依赖 Wallis 公式的方法。
记
n n n R R R V k ( R ) V_k(R) V k ( R ) S k ( R ) S_k(R) S k ( R ) V k ( R ) = ∫ − R R V k − 1 ( R 2 − r 2 ) d r = ∫ − π 2 π 2 V k − 1 ( R cos θ ) d ( R sin θ ) = R ∫ − π 2 π 2 V k − 1 ( R cos θ ) cos θ d θ S k ( R ) = ∫ − π 2 π 2 S k − 1 ( R cos θ ) R d θ = R ∫ − π 2 π 2 S k − 1 ( R cos θ ) d θ V k ( R ) = ∫ 0 R S k ( r ) d r
\begin{aligned}
V_k(R)&=\int_{-R}^RV_{k-1}(\sqrt{R^2-r^2})\,\mathrm{d}r\\
&=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}V_{k-1}(R\cos\theta)\,\mathrm{d}(R\sin\theta)\\
&=R\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}V_{k-1}(R\cos\theta)\cos\theta\,\mathrm{d}\theta\\
S_k(R)&=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}S_{k-1}(R\cos\theta)R\,\mathrm{d}\theta\\
&=R\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}S_{k-1}(R\cos\theta)\,\mathrm{d}\theta\\
V_k(R)&=\int_{0}^RS_k(r)\,\mathrm{d}r
\end{aligned}
V k ( R ) S k ( R ) V k ( R ) = ∫ − R R V k − 1 ( R 2 − r 2 ) d r = ∫ − 2 π 2 π V k − 1 ( R cos θ ) d ( R sin θ ) = R ∫ − 2 π 2 π V k − 1 ( R cos θ ) cos θ d θ = ∫ − 2 π 2 π S k − 1 ( R cos θ ) R d θ = R ∫ − 2 π 2 π S k − 1 ( R cos θ ) d θ = ∫ 0 R S k ( r ) d r
并且我们知道
V 1 ( R ) = 2 R , S 1 ( R ) = 2 , V 2 ( R ) = π R 2 , S 2 ( R ) = 2 π R , V 3 ( R ) = 4 3 π R 3 , S 3 ( R ) = 4 π R 2 V_1(R)=2R,S_1(R)=2,V_2(R)=\pi R^2,S_2(R)=2\pi R,V_3(R)=\frac{4}{3}\pi R^3,S_3(R)=4\pi R^2 V 1 ( R ) = 2 R , S 1 ( R ) = 2 , V 2 ( R ) = π R 2 , S 2 ( R ) = 2 π R , V 3 ( R ) = 3 4 π R 3 , S 3 ( R ) = 4 π R 2 V k ( R ) = a k R k , S k ( R ) = b k R k − 1 V_k(R)=a_kR^k,S_k(R)=b_kR^{k-1} V k ( R ) = a k R k , S k ( R ) = b k R k − 1 V k ( R ) = R ∫ − π 2 π 2 V k − 1 ( R cos θ ) cos θ d θ = R ∫ − π 2 π 2 a k − 1 R k − 1 cos k θ d θ = R k a k − 1 ∫ − π 2 π 2 cos k θ d θ S k ( R ) = R ∫ − π 2 π 2 S k − 1 ( R cos θ ) d θ = R ∫ − π 2 π 2 b k − 1 R k − 2 cos k − 2 θ d θ = R k − 1 b k − 1 ∫ − π 2 π 2 cos k − 2 θ d θ V k ( R ) = ∫ 0 R S k ( r ) d r = ∫ 0 R b k r k − 1 d r = 1 k b k R k
\begin{aligned}
V_k(R)&=R\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}V_{k-1}(R\cos\theta)\cos\theta\,\mathrm{d}\theta\\
&=R\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}a_{k-1}R^{k-1}\cos^k\theta\,\mathrm{d}\theta\\
&=R^ka_{k-1}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^k\theta\,\mathrm{d}\theta\\
S_k(R)&=R\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}S_{k-1}(R\cos\theta)\,\mathrm{d}\theta\\
&=R\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}b_{k-1}R^{k-2}\cos^{k-2}\theta\,\mathrm{d}\theta\\
&=R_{k-1}b_{k-1}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^{k-2}\theta\,\mathrm{d}\theta\\
V_k(R)&=\int_{0}^RS_k(r)\,\mathrm{d}r\\
&=\int_0^Rb_kr^{k-1}\,\mathrm{d}r\\
&=\frac{1}{k}b_kR^k
\end{aligned}
V k ( R ) S k ( R ) V k ( R ) = R ∫ − 2 π 2 π V k − 1 ( R cos θ ) cos θ d θ = R ∫ − 2 π 2 π a k − 1 R k − 1 cos k θ d θ = R k a k − 1 ∫ − 2 π 2 π cos k θ d θ = R ∫ − 2 π 2 π S k − 1 ( R cos θ ) d θ = R ∫ − 2 π 2 π b k − 1 R k − 2 cos k − 2 θ d θ = R k − 1 b k − 1 ∫ − 2 π 2 π cos k − 2 θ d θ = ∫ 0 R S k ( r ) d r = ∫ 0 R b k r k − 1 d r = k 1 b k R k
记
f ( k ) = ∫ − π 2 π 2 cos k θ d θ f(k)=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\cos^k\theta\,\mathrm{d}\theta f ( k ) = ∫ − 2 π 2 π cos k θ d θ a k = f ( k ) a k − 1 b k = f ( k − 2 ) b k − 1 a k = 1 k b k
\begin{align}
a_k&=f(k)a_{k-1}\\
b_k&=f(k-2)b_{k-1}\\
a_k&=\frac{1}{k}b_k
\end{align}
a k b k a k = f ( k ) a k − 1 = f ( k − 2 ) b k − 1 = k 1 b k a 1 = b 1 = 2 , f ( 0 ) = π , f ( 1 ) = 2 a_1=b_1=2,f(0)=\pi,f(1)=2 a 1 = b 1 = 2 , f ( 0 ) = π , f ( 1 ) = 2 ( 1 ) ( 2 ) (1)(2) ( 1 ) ( 2 ) a k = 2 f ( 2 ) f ( 3 ) ⋯ f ( k ) b k = 2 f ( 0 ) f ( 1 ) ⋯ f ( k − 2 )
\begin{aligned}
&a_k=2f(2)f(3)\cdots f(k)\\
&b_k=2f(0)f(1)\cdots f(k-2)
\end{aligned}
a k = 2 f ( 2 ) f ( 3 ) ⋯ f ( k ) b k = 2 f ( 0 ) f ( 1 ) ⋯ f ( k − 2 ) ( 3 ) (3) ( 3 ) f ( k − 1 ) f ( k ) = 2 π k
f(k-1)f(k)=\frac{2\pi}{k}
f ( k − 1 ) f ( k ) = k 2 π f ( 0 ) = π , f ( 2 ) = π 2 f(0)=\pi,f(2)=\frac{\pi}{2} f ( 0 ) = π , f ( 2 ) = 2 π ( 1 ) ( 2 ) (1)(2) ( 1 ) ( 2 ) V k ( R ) = { π k / 2 ( k / 2 ) ! R k k ≡ 0 ( m o d 2 ) π ( k − 1 ) / 2 1 2 3 2 ⋯ k 2 R k = π k / 2 ( − 1 2 ) ! 1 2 3 2 ⋯ k 2 R k k ≡ 1 ( m o d 2 ) = π k / 2 ( k / 2 ) ! R k S k ( R ) = V k ( R ) k R
\begin{aligned}
V_k(R)&=\begin{cases}
\frac{\pi^{k/2}}{(k/2)!}R^k & k\equiv0\pmod 2\\
\frac{\pi^{(k-1)/2}}{\frac{1}{2}\frac{3}{2}\cdots\frac{k}{2}}R^k=\frac{\pi^{k/2}}{(-\frac{1}{2})!\frac{1}{2}\frac{3}{2}\cdots\frac{k}{2}}R^k & k\equiv1\pmod 2
\end{cases}\\
&=\frac{\pi^{k/2}}{(k/2)!}R^k\\
S_k(R)&=\frac{V_k(R)}{kR}
\end{aligned}
V k ( R ) S k ( R ) = { ( k /2 )! π k /2 R k 2 1 2 3 ⋯ 2 k π ( k − 1 ) /2 R k = ( − 2 1 )! 2 1 2 3 ⋯ 2 k π k /2 R k k ≡ 0 ( mod 2 ) k ≡ 1 ( mod 2 ) = ( k /2 )! π k /2 R k = k R V k ( R )