题目
18.求 e^1-1(pi)/(2), exp[(1+ipi)/4], 3^i 和 (1+i)^i 的值.
18.求 $e^{1-1\frac{\pi}{2}}$, $exp[(1+i\pi)/4]$, $ 3^{i}$ 和 $(1+i)^{i}$ 的值.
题目解答
答案
1. $e^{1-i\frac{\pi}{2}} = e \cdot e^{-i\frac{\pi}{2}} = e \cdot (-i) = -ie$
2. $e^{\frac{1+i\pi}{4}} = e^{\frac{1}{4}} \cdot e^{i\frac{\pi}{4}} = \frac{\sqrt{2}}{2} e^{\frac{1}{4}} (1 + i)$
3. $3^i = e^{i \ln 3} = e^{-2k\pi} (\cos(\ln 3) + i \sin(\ln 3))$，主值为 $\cos(\ln 3) + i \sin(\ln 3)$
4. $(1+i)^i = e^{i \ln(1+i)} = e^{-(\frac{\pi}{4} + 2k\pi)} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right]$，主值为 $e^{-\frac{\pi}{4}} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right]$
\[
\boxed{
\begin{array}{ll}
1. & -ie \\
2. & \frac{\sqrt{2}}{2} e^{\frac{1}{4}} (1 + i) \\
3. & e^{-2k\pi} (\cos(\ln 3) + i \sin(\ln 3)), \text{主值: } \cos(\ln 3) + i \sin(\ln 3) \\
4. & e^{-(\frac{\pi}{4} + 2k\pi)} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right], \text{主值: } e^{-\frac{\pi}{4}} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right] \\
\end{array}
}
\]
解析
步骤 1：计算 $e^{1-i\frac{\pi}{2}}$
$e^{1-i\frac{\pi}{2}} = e \cdot e^{-i\frac{\pi}{2}} = e \cdot (-i) = -ie$
步骤 2：计算 $exp[(1+i\pi)/4]$
$exp[(1+i\pi)/4] = e^{\frac{1}{4}} \cdot e^{i\frac{\pi}{4}} = \frac{\sqrt{2}}{2} e^{\frac{1}{4}} (1 + i)$
步骤 3：计算 $3^{i}$
$3^i = e^{i \ln 3} = e^{-2k\pi} (\cos(\ln 3) + i \sin(\ln 3))$，主值为 $\cos(\ln 3) + i \sin(\ln 3)$
步骤 4：计算 $(1+i)^{i}$
$(1+i)^i = e^{i \ln(1+i)} = e^{-(\frac{\pi}{4} + 2k\pi)} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right]$，主值为 $e^{-\frac{\pi}{4}} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right]$
$e^{1-i\frac{\pi}{2}} = e \cdot e^{-i\frac{\pi}{2}} = e \cdot (-i) = -ie$
步骤 2：计算 $exp[(1+i\pi)/4]$
$exp[(1+i\pi)/4] = e^{\frac{1}{4}} \cdot e^{i\frac{\pi}{4}} = \frac{\sqrt{2}}{2} e^{\frac{1}{4}} (1 + i)$
步骤 3：计算 $3^{i}$
$3^i = e^{i \ln 3} = e^{-2k\pi} (\cos(\ln 3) + i \sin(\ln 3))$，主值为 $\cos(\ln 3) + i \sin(\ln 3)$
步骤 4：计算 $(1+i)^{i}$
$(1+i)^i = e^{i \ln(1+i)} = e^{-(\frac{\pi}{4} + 2k\pi)} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right]$，主值为 $e^{-\frac{\pi}{4}} \left[\cos\left(\frac{\ln 2}{2}\right) + i \sin\left(\frac{\ln 2}{2}\right)\right]$