Line Integrals

Reading

Section 4.1

Problems

Practice problems (page 56): 1, 2, 3, 4

Topics to know

1. Notions of piecewise-differentiable curves and of smooth curves (4.2)
2. Integral of a complex-valued function of a real variable (4.1).
3. Integrate \(1/z\) and \(1/z^2\) over a circle around \(0\). (page 48)
4. Integral of \(f(z)\) along a smooth curve.
5. Integrals of \(f(z)\) over equivalent curves are equal.
6. Integral over the opposite curve is the negative of the integral over the curve.
7. Linearity of the integral (proposition 4.8).
8. \(\displaystyle\int_a^b G(t)dt \ll \int_a^b |G(t)|dt\).
   • \(\int_a^b G(t) dt = Re^{i\theta}\)
   • \(\int_a^b e^{i\theta} G(t) dt = R\) is real
   • \(e^{i\theta} G(t) = A(t) + i B(t)\)
   • Then \(R = \int_a^b A(t)dt = \int_a^b \textrm{Re}\left(e^{-\theta}G(t)\right)dt\leq \int_a^b |G(t)|dt\).
9. ML-formula: If \(f\ll M\) along a curve \(C\) of length \(L\), then: \[\int_C f(z)dz = \int_a^b f(z(t))\dot z(t)\ll \int_a^b|f(z(t))\dot z(t)|dt\ll M\int_a^b|\dot z(t)|dt = ML\]
10. If \(f_n\to f\) uniformly on \(C\), then \(\int_C f_n(z)dz\to \int_C f(z)dz\).
    • Notion of uniform convergence: For every \(\epsilon > 0\) there is an \(N\) such that for all \(n\geq N\) and for all \(x\in C\) we have \(|f_n(x) - f(x)|<\epsilon\).
    • Key idea: \(n\) depends only on $epsilon, but works for all \(x\).
11. If \(F'(z) = f(z)\) then \(\int_C f(z)dz = F(z(b)) - F(z(a))\).
    • If \(\lambda(t)\) is the curve, \(\gamma(t) = F(\lambda(t))\)
    • Then \(\dot\gamma(t) = F'(\lambda(t))\dot\lambda(t) = f(\lambda(t))\dot\lambda(t)\)
    • \(\int_C f(z)dz = \int_a^b f(\lambda(t))\dot\lambda(t)dt = \int_a^b \dot\gamma(t) = \gamma(b) - \gamma(a) = F(z(b)) - F(z(a))\)