Evaluate the contour integral $\oint_C \frac{z^2 + 1}{z(z - 2)} dz$ counter-clockwise around the circle $C: |z| = 1$. (Give the answer as a scalar multiple of $\pi i$, i.e. if the integral is $k \pi i$, enter the value of $k$)

Answer Key:

-1.0

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Identification of Singularities:** The integrand $f(z) = \frac{z^2 + 1}{z(z - 2)}$ has simple poles at $z = 0$ and $z = 2$. 2. **Stage 2: Contour Boundary Condition:** The contour is $|z| = 1$ (circle of radius 1 centered at origin). * $z = 0$ lies strictly INSIDE the contour ($|0| < 1$). * $z = 2$ lies strictly OUTSIDE the contour ($|2| > 1$). 3. **Stage 3: Residue Calculation at $z = 0$:** * $\text{Res}(f, 0) = \lim_{z\to 0} [z \cdot f(z)] = \lim_{z\to 0} \frac{z^2 + 1}{z - 2} = \frac{0^2 + 1}{0 - 2} = -\frac{1}{2}$. 4. **Stage 4: Cauchy Residue Integration:** * $\oint_C f(z) dz = 2\pi i \times \text{Res}(f, 0) = 2\pi i \left(-\frac{1}{2}\right) = -\pi i = -1.0 \pi i$. * Therefore, scalar $k = -1.0$. 5. **Stage 5: Official Reference:** Erwin Kreyszig, *Advanced Engineering Mathematics*, Chapter 16.

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