Latihan Soal Bab 4: Pengintegralan Kompleks

Pembahasan: Gunakan Rumus Turunan Cauchy dengan $f(z)=z^3+2z,z_0=i,n=2$: $f^{\prime }(z)=3z^2+2⟹f^{\prime \prime }(z)=6z⟹f^{\prime \prime }(i)=6i$. ${\oint }_C\frac{z^3+2z}{(z-i{)}^3}dz=\frac{2\pi i}{2!}{f}^{\prime \prime }(i)=\pi i(6i)=-6\pi$. $■$

Pembahasan: Tinjau integral ${\oint }_{|z|=1}\frac{e^z}{z}dz$.

• Sisi 1 (Cauchy): Titik $z=0$ di dalam $|z|=1$, $f(z)=e^z⟹\oint \frac{e^z}{z}dz=2\pi i{e}^0=2\pi i$.
• Sisi 2 (Parameterisasi $z=e^{i\theta }$): ${\int }_0^{2\pi }\frac{e^{\cos \theta +i\sin \theta }}{e^{i\theta }}i{e}^{i\theta }d\theta =i{\int }_0^{2\pi }{e}^{\cos \theta }(\cos (\sin \theta )+i\sin (\sin \theta ))d\theta$. Samakan bagian imajiner: ${\int }_0^{2\pi }{e}^{\cos \theta }\cos (\sin \theta )d\theta =2\pi$. $■$