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受欢迎的代数 >

x^x=pi

解答

x^{x} = π

解答

x = \frac{ln ( π )}{W _{0} ( ln ( π ) )}

+1

十进制

x = 1.85410 \dots

求解步骤

x^{x} = π

将

x^{x} = π

调整为朗伯形式:

x e^{- \frac{l n ( π )}{x}} = 1

用

\frac{ln ( π )}{x} = u

和

x = \frac{ln ( π )}{u}

改写方程式

(\frac{ln ( π )}{u}) e^{- u} = 1

改写

(\frac{ln ( π )}{u}) e^{- u} = 1

为朗伯形式:

u e^{u} = ln (π)

解

u e^{u} = ln (π) : u = W_{0} (ln (π))

验证解:

u = W_{0} (ln (π))

真

解是

u = W_{0} (ln (π))

代回

u = \frac{ln ( π )}{x}

，求解

x

解

\frac{ln ( π )}{x} = W_{0} (ln (π)) : x = \frac{ln ( π )}{W _{0} ( ln ( π ) )}

x = \frac{ln ( π )}{W _{0} ( ln ( π ) )}

作图

流行的例子

40=100e^{-0.03t}

40 = 100 e^{- 0.03 t}

solvefor x,y=e^{x+y}

so l v e f or x, y = e^{x + y}

solvefor k, n/(2^k)=0

so l v e f or k, \frac{n}{2 ^{k}} = 0

3 \cdot 2^{x} = 6^{x}

solvefor t,y=e^{2t}

so l v e f or t, y = e^{2 t}