Conversely, suppose that \(e: I \to \mathbb{R}\) is a primitive of \(f.\) The difference \(d\) between \(e\) and the function \[ g: x \in I \mapsto e(a) + \int_a^x f(t) \, dt \] is zero at \(a\) and has a zero derivative on \(I.\) By the mean value theorem, for any \(x \in I\) such that \(x\neq a,\) there is a \(b \in I\) such that \[ \frac{d(x) - d(a)}{x - a} = d'(b) = 0, \] hence \(d(x) = d(a) = 0\) and therefore \(e = g.\) ‌\(\blacksquare\)

Similarly, \[ \begin{split} \int_{\gamma} f(z) dz &= \int_0^1 f(\gamma(t)) \gamma'(t) \, dt \\ &= \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f(\gamma(t)) \gamma'(t) \, dt \\ &= \sum_{k=0}^{n-1} \int_{t_k}^{t_{k+1}} f\left(\gamma_k\left( \frac{t-t_k}{t_{k+1} - t_k} \right) \right) \frac{\gamma_k'\left( \frac{t-t_k}{t_{k+1} - t_k} \right)}{t_k - t_{k-1}} \, dt \\ &= \sum_{k=1}^n \int_0^1 f(\gamma_k(t)) \gamma_k'(t) \, dt \\ &= \sum_{k=1}^n \int_{\gamma_k} f(z) \, dz \end{split} \] ‌\(\blacksquare\)

• The path \(\mu = \gamma \circ \phi\) is a continuously differentiable path.

• It has the same initial point, terminal point and image as \(\gamma.\)

• The length of \(\mu\) and \(\gamma\) are identical.

• For any continuous function \(f: \gamma([0,1]) \to \mathbb{C},\) \[ \int_{\mu} f(z) \, dz = \int_{\gamma} f(z) \, dz. \]

The length of \(\mu\) is \[ \ell(\mu) = \int_0^1 |\mu'(t)|\, dt = \int_{0}^1 |\gamma' (\phi(t)) \phi'(t)| \, dt = \int_{0}^1 |\gamma' (\phi(t))| \, \phi'(t) dt \] The change of variable \(s = \phi(t)\) provides \[ \int_{0}^1 |\gamma' (\phi(t))| \, \phi'(t) dt = \int_0^1 |\gamma'(s)| \, ds, \] hence the lengths of \(\gamma\) and \(\mu\) are equal. We also have \[ \int_{\mu} f(z) \, dz = \int_{0}^1 (f \circ \mu)(t) \mu'(t)\,dt = \int_{0}^1 (f \circ \gamma)(\phi(t)) \gamma'(\phi(t))\,(\phi'(t) dt). \] The same change of variable leads to \[ \int_{\mu} f(z) \, dz = \int_{0}^1 (f \circ \gamma)(s) \gamma'(s)\,ds = \int_{\gamma} f(z) \, dz, \] which concludes the proof. ‌\(\blacksquare\)

Moreover, \[ \begin{split} \int_{\gamma} g(f(w)) f'(w) \, dw = &\, \int_0^1 g(f(\gamma(t)) f'(\gamma(t)) \gamma'(t) \, dt \\ = &\, \int_0^1 g(f(\gamma(t)) (f \circ \gamma)'(t) \, dt \\ = &\, \int_{f \circ \gamma} g(w) \, dw \end{split} \] ‌\(\blacksquare\)

Rectifiable Paths

The definition we used so far for “rectifiable” is a conservative one. In this section, we come up with a more general definition of the concept that still meets the requirements for the definition of line integrals.

To “rectify” a path (from Latin rectus “straight” and facere “to make”) is to straighten – or by extension to compute its length, which is a trivial operation once a path has been straightened.

The general definition of the length of a path does not require line integrals. Instead, consider any partition \((t_0,\dots,t_n)\) of the interval \([0,1]\) and the path \(\mu= \mu_1 \, |_{t_1} \, \dots \,|_{t_{n-1}} \mu_n\) where \[ \mu_{k}(t) = (1-t) \gamma(t_{k-1}) + t \gamma(t_{k}). \] We may define the length of such a combination of straight lines as \[ \ell(\mu) = \sum_{k=1}^{n-1} |\gamma(t_{k+1}) - \gamma(t_{k})|. \] As the straight line is the shortest path between two points, this number should provide a lower bound of the length of \(\gamma.\) On the other hand, using finer partitions of the interval \([0,1]\) should also provide better approximations of the length of \(\gamma.\) Following this idea, we may define the length of \(\gamma\) as the supremum of the length of \(\mu\) for all possible partitions of \([0,1]\): \[ \ell(\gamma) = \sup \left\{ \sum_{k=0}^{n-1} |\gamma(t_{k+1}) - \gamma(t_{k})| \; \left| \; \vphantom{\sum_k^k} n\in \mathbb{N}^*, \, t_0=0 < \dots < t_n =1 \right. \right\} \]

Not every path has a finite length; those who have are by definition rectifiable. In general, a function \(\gamma:[0,1] \mapsto \mathbb{C}\) whose length is finite – even if it is not continuous – is of bounded variation.

The Line Integral

To define line integrals along the path \(\gamma,\) it is enough that \(\gamma\) is of bounded variation. For any such function \(\gamma,\) we may build a (complex-valued, Borel) measure on \([0,1]\) denoted \(d\gamma.\) This measure is defined by its integral of any continuous function \(\phi:[0,1]\to \mathbb{C},\) as a limit of Riemann(-Stieltjes) sums \[ \int_{[0,1]} \phi \, d\gamma = \lim \sum_{m=0}^{n-1} \phi (t_m) (\gamma(t_{m+1}) - \gamma(t_m)). \] The limit is taken over the partitions of the interval \([0,1]\) with \[ \max \, \{|t_{m+1} - t_{m}| \; | \; m \in \{0,\dots, n-1\}\} \to 0. \] The line integral of a continuous function \(f: \gamma([0,1]) \to \mathbb{C}\) is then defined by \[ \int_{\gamma} f(z) \, dz = \int_{[0,1]} (f \circ \gamma) \, d\gamma. \] The total variation \(|d\gamma|\) of \(d\gamma\) is the positive measure defined by \[ |d\gamma|(A) = \sup_{\mathfrak{P}} \sum_{B \in \mathfrak{P}} |d\gamma(B)| \] where the supremum is taken over all finite partitions \(\mathfrak{P}\) of \(A\) into measurable sets. This measure provides an integral expression for the length of \(\gamma\): \[ \ell(\gamma) = \int_{[0,1]} |d\gamma|. \]

A Non-Rectifiable Curve

The Koch snowflake (Koch 1904) is an example of a continuous curve which is is nowhere differentiable; it is also a non-rectifiable closed path. It is defined as the limit of a sequence of polylines \(\gamma_n.\) The first element of this sequence is an oriented equilateral triangle: \[ \gamma_1 = [0 \to 1 \to e^{i\pi3} \to 0]. \] Then, \(\gamma_{n+1}\) is defined as a transformation of \(\gamma_n\): every oriented line segment \([a \to a+h]\) that composes \(\gamma_n\) is replaced by the polyline: \[ \left[a \to a + \frac{h}{3} \to a + \left(1+e^{-i\pi/3} \right)\frac{h}{3}\to a + 2\frac{h}{3} \to a+h\right] \]

The Koch snowflake \(\gamma\) is defined as the limit of the \(\gamma_n\) sequence. The geometric construction yields that for any \(n\) greater than zero, \[ \forall \, t \in [0,1], \; |\gamma_{n+1}(t) - \gamma_n(t)| \leq \left(\frac{1}{3}\right)^{n} \frac{\sqrt{3}}{2}. \] As \(\sum_{p=0}^{+\infty} \left( \frac{1}{3}\right)^p = \frac{1}{1-1/3} = \frac{3}{2},\) for any positive integer \(p\) we have \[ \forall \, t \in [0,1], \; |\gamma_{n+p}(t) - \gamma_n(t)| \leq \left(\frac{1}{3}\right)^{n} \frac{3}{2}\frac{\sqrt{3}}{2}. \] The sequence \(\gamma_n\) is a Cauchy sequence in the space of continuous and complex-valued functions defined on \([0,1];\) its uniform limit exists and is also continuous.

On the other hand, the curve is not rectifiable. First, the definition of the sequence \(\gamma_n\) makes it plain that every iteration increases the initial length of the path by one-third: \[ \ell(\gamma_n) = 3 \times \left(\frac{4}{3}\right)^{n-1}. \] The length of \(\gamma_n\) tends to \(+\infty\) when \(n \to +\infty.\) Now, every point at the junction of the segments of the polyline \(\gamma_n\) also belongs to the Koch snowflake; more precisely \[ \forall \, m \in \{0,\dots, 3 \times4^{n-1}\}, \; \gamma\left(\frac{m}{3 \times 4^{n-1}}\right) = \gamma_n\left(\frac{m}{3 \times 4^{n-1}}\right). \] Therefore \[ \ell(\gamma) \geq \sum_{m=0}^{3 \times 4^{n-1} -1} \left| \gamma\left(\frac{m+1}{3 \times 4^{n-1}}\right) - \gamma\left(\frac{m}{3 \times 4^{n-1}}\right)\right| = \ell(\gamma_n) \] and thus \(\ell(\gamma) = +\infty\): the path \(\gamma\) is not rectifiable.