6.4 參考解答

  1. ∫czez dz=∫−1−iπ22+iπzez dz=zez ∣−1−iπ22+iπ−∫−1−iπ22+iπez dz=(zez−ez) ∣−1−iπ22+iπ\int_c ze^z\space dz=\int_{-1-i\frac{\pi}{2}}^{2+i\pi} ze^z\space dz=ze^z\space |_{-1-i\frac{\pi}{2}}^{2+i\pi}-\int_{-1-i\frac{\pi}{2}}^{2+i\pi} e^z\space dz=(ze^z-e^z)\space |_{-1-i\frac{\pi}{2}}^{2+i\pi}

    =(2+iπ−1)e2+iπ−(−1−iπ2−1)e−1−iπ2=(2+i\pi-1)e^{2+i\pi}-(-1-i\frac{\pi}{2}-1)e^{-1-i\frac{\pi}{2}}

    =(1+iπ)e2+iπ+(2+iπ2)e−1−iπ2=(1+i\pi)e^{2+i\pi}+(2+i\frac{\pi}{2})e^{-1-i\frac{\pi}{2}}

    =(1+iπ)e2(cos⁡π+isin⁡π)+(2+iπ2)e−1(cos⁡π2−isin⁡π2)=(1+i\pi)e^2(\cos{\pi}+i\sin{\pi})+(2+i\frac{\pi}{2})e^{-1}(\cos{\frac{\pi}{2}}-i\sin{\frac{\pi}{2}})

    =(−1−iπ)e2+(−i)(2+iπ2)e−1=(-1-i\pi)e^2+(-i)(2+i\frac{\pi}{2})e^{-1}

    =−e2+π2e−1+i(−πe2−2e−1)=-e^2+\frac{\pi}{2}e^{-1}+i(-\pi e^2-2e^{-1})

    =(−1+12e−3π)e2+i(−π−2e−3)e2=(-1+\frac{1}{2}e^{-3}\pi)e^2+i(-\pi-2e^{-3})e^2

9. ∫Czcos⁡z dz=∫0izcos⁡z dz=zsin⁡z ∣0i−∫0isin⁡z dz\int_C z\cos{z}\space dz=\int_{0}^{i} z\cos{z}\space dz=z\sin{z}\space |_{0}^{i}-\int_{0}^{i} \sin{z}\space dz

=isin⁡i−(−cos⁡z) ∣0i=i\sin{i}-(-\cos{z})\space |_{0}^{i}

=isin⁡i−(−cos⁡i+cos⁡0)=i\sin{i}-(-\cos{i}+\cos{0})

=isin⁡i+cos⁡i−1=i\sin{i}+\cos{i}-1

=−1−sinh⁡1+cosh⁡1=-1-\sinh{1}+\cosh{1}

  1. ∫C2z−1z2−z dz=∫C1z dz+∫C1z−1 dz\int_C \frac{2z-1}{z^2-z}\space dz=\int_C \frac{1}{z}\space dz+\int_C \frac{1}{z-1}\space dz

    =∫22+i1z dz+∫22+i1z−1 dz=\int_{2}^{2+i} \frac{1}{z}\space dz+\int_{2}^{2+i} \frac{1}{z-1}\space dz

    =[Log(2+i)−Log(2)]+[Log(2+i−1)−Log(2−1)]=[\text{Log}(2+i)-\text{Log}(2)]+[\text{Log}(2+i-1)-\text{Log}(2-1)]

    =[Log(2+i)−Log(2)]+[Log(1+i)−Log(1)]=[\text{Log}(2+i)-\text{Log}(2)]+[\text{Log}(1+i)-\text{Log}(1)]

    =(ln⁡5+tan⁡−112)−(ln⁡2+i⋅0)+(ln⁡2+iπ4)−(ln⁡1+i⋅0)=(\ln{\sqrt{5}}+\tan^{-1}\frac{1}{2})-(\ln{2}+i \cdot 0)+(\ln{\sqrt{2}+i\frac{\pi}{4}})-(\ln{1}+i\cdot 0)

    =ln⁡5−ln⁡2+ln⁡2+i(π4+tan⁡−112)=\ln{\sqrt{5}}-\ln{2}+\ln{\sqrt{2}}+i(\frac{\pi}{4}+\tan^{-1}\frac{1}{2})

    =ln⁡5−ln⁡2+i(π4+tan⁡−112)=\ln{\sqrt{5}}-\ln{\sqrt{2}}+i(\frac{\pi}{4}+\tan^{-1}\frac{1}{2})