\(f: D\rightarrow {\mathbb C}\) be a continuous function, where \(D\subset {\mathbb C}\) is a domain.Let \(\alpha:[a,b]\rightarrow D\) be a smooth curve. Give a precise definition of the complex line integral \[\int_{\alpha} f.\]
Assume that there exists a constant \(M\) such that \(|f(\tau)|\leq M\) for all \(\tau\in \mbox{\textrm Image}(\alpha\)). Prove that \[\big | \int_{\alpha} f \big |\leq M \times \mbox{\textrm length}(\alpha).\]
Let \(C_R\) be the circle \(|z|=R\), described in the counterclockwise direction, where \(R>1\). Provide an upper bound for \(\big | \int_{C_R} \dfrac{\log{(z)} }{z^2} \big |,\) which depends only on \(R\) and other constants.
Solution
Goal:
State the precise definition of the complex line integral \(\int_\alpha f(z)\,dz\).
Prove the standard \(ML\)-inequality \(\left|\int_\alpha f\right| \leq M \cdot \text{length}(\alpha)\).
Provide an explicit upper bound for \(\left|\int_{C_R} \frac{\log z}{z^2}\,dz\right|\) for \(R > 1\) using a single-valued branch on the contour (except at the branch cut point) or along the circle.
Part (a): Definition of Complex Line Integral
1. Definition of \(\int_\alpha f(z)\,dz\).
1.1. Let \(\alpha: [a, b] \to D \subset \mathbb{C}\) be a \(C^1\) (smooth) parametrization, \(\alpha(t) = x(t) + i y(t)\), and let \(f: D \to \mathbb{C}\) be continuous.Proof: Setting hypotheses.
1.2. The complex line integral of \(f\) along \(\alpha\) is defined by the Riemann integral: \[\int_\alpha f(z) \, dz \coloneqq \int_a^b f(\alpha(t)) \alpha'(t) \, dt.\] Proof: Standard definition of contour integral in complex analysis.
1.3. Writing \(f = u+iv\) and \(\alpha' = x'+iy'\), this corresponds to the pair of real Riemann integrals \(\int_a^b (u x' - v y')\,dt + i \int_a^b (u y' + v x')\,dt\).Proof: Algebraic expansion of complex multiplication.
1.4. Q.E.D.
Part (b): Proof of the \(ML\)-Inequality
2. Let \(I = \int_\alpha f(z)\,dz\). If \(I = 0\), then \(|I| = 0 \leq M \cdot \text{length}(\alpha)\) because \(M \geq 0\) and \(\text{length}(\alpha) \geq 0\).
2.1. If \(I = 0\), then \(|I| = 0 \leq M \cdot \text{length}(\alpha)\) because \(M \geq 0\) and \(\text{length}(\alpha) \geq 0\).Proof: Non-negativity of length and supremum norm.
2.2. Q.E.D.
3. Suppose \(I \neq 0\). Write \(I = |I| e^{i\theta}\) for some \(\theta \in \mathbb{R}\).
3.1. Then \(|I| = e^{-i\theta} I = e^{-i\theta} \int_a^b f(\alpha(t)) \alpha'(t) \, dt = \int_a^b e^{-i\theta} f(\alpha(t)) \alpha'(t) \, dt\).Proof: Linearity of the integral with respect to complex scalars.
3.2. Since \(|I|\) is a positive real number, \(|I| = \text{Re}(|I|) = \int_a^b \text{Re}\left( e^{-i\theta} f(\alpha(t)) \alpha'(t) \right) dt\).Proof: Taking real parts.
3.3. For any complex number \(w \in \mathbb{C}\), \(\text{Re}(w) \leq |w|\).Applying this to \(w = e^{-i\theta} f(\alpha(t)) \alpha'(t)\): \[\text{Re}\left( e^{-i\theta} f(\alpha(t)) \alpha'(t) \right) \leq \left| e^{-i\theta} f(\alpha(t)) \alpha'(t) \right| = |e^{-i\theta}| |f(\alpha(t))| |\alpha'(t)| = |f(\alpha(t))| |\alpha'(t)|.\] Proof: \(|e^{-i\theta}| = 1\) and multiplicativity of complex modulus.
3.4. By hypothesis, \(|f(\alpha(t))| \leq M\) for all \(t \in [a, b]\).Proof: Given bound on \(\text{Image}(\alpha)\).
3.5. Integrating the inequality: \[|I| \leq \int_a^b |f(\alpha(t))| |\alpha'(t)| \, dt \leq \int_a^b M |\alpha'(t)| \, dt = M \int_a^b |\alpha'(t)| \, dt.\] Proof: Monotonicity of the Riemann integral for real-valued functions.
3.6. By definition, the arc length of the smooth curve \(\alpha\) is \(\text{length}(\alpha) = \int_a^b |\alpha'(t)| \, dt\).Proof: Standard definition of curve length.
3.7. Therefore, \(\left| \int_\alpha f \right| \leq M \cdot \text{length}(\alpha)\).Proof: Substitution of 6 into
3.5.
3.8. Q.E.D.
Part (c): Upper Bound for \(\left|\int_{C_R} \frac{\log z}{z^2}\,dz\right|\)
4. Bound the integrand on the circle \(C_R: z = R e^{i\theta}\) (\(\theta \in [-\pi, \pi]\)).
4.1. On \(C_R\), \(|z| = R > 1\), so \(|z^2| = R^2\).Proof: Modulus of power.
4.2. Using the standard branch of \(\log z = \ln|z| + i \arg(z) = \ln R + i\theta\) with \(\theta \in [-\pi, \pi]\): \[|\log z| = |\ln R + i\theta| = \sqrt{(\ln R)^2 + \theta^2} \leq \sqrt{(\ln R)^2 + \pi^2} \leq \ln R + \pi.\] Proof: Pythagorean modulus and the triangle inequality \(\sqrt{a^2+b^2} \leq |a|+|b|\).
4.3. Therefore, for all \(z \in C_R\): \[\left| \frac{\log z}{z^2} \right| \leq \frac{\sqrt{(\ln R)^2 + \pi^2}}{R^2} \leq \frac{\ln R + \pi}{R^2} \eqqcolon M.\] Proof: Quotient of upper bound on numerator by exact denominator modulus.
4.4. The length of the circle \(C_R\) is \(L = 2\pi R\).Proof: Circumference of a circle of radius \(R\).
4.5. By the \(ML\)-inequality from Part (b): \[\left| \int_{C_R} \frac{\log z}{z^2} \, dz \right| \leq M \cdot L = \frac{\sqrt{(\ln R)^2 + \pi^2}}{R^2} \cdot 2\pi R = \frac{2\pi \sqrt{(\ln R)^2 + \pi^2}}{R} \leq \frac{2\pi(\ln R + \pi)}{R}.\] Proof: Direct application of <1>3.<2>7.
4.6. Q.E.D.