# Exercises

## Primitives of Power Functions

### Question

Determine the primitives of the power $$z \mapsto z^n$$ – defined on $$\mathbb{C}$$ if $$n$$ nonnegative and on $$\mathbb{C}^*$$ otherwise – or prove that no such function exist.

### Answer

If $$n \neq -1$$, the function $$z \mapsto z^{n+1}/(n+1)$$ is a primitive of $$z \mapsto z^n$$. As $$\mathbb{C}$$ and $$\mathbb{C}^*$$ are path-connected, the other primitives differ from this one by a constant.

If $$n=-1$$, no primitive exist: the function $$\gamma: t \in [0,1] \to e^{i2\pi t}$$ is a closed rectifiable path of $$\mathbb{C}^*$$ and $\int_{\gamma} \frac{dz}{z} = \int_0^1 \frac{e^{i2\pi t} i2\pi}{e^{i2 \pi t}} \, dt = i2\pi,$ which is nonzero.

## Primitive of a Rational Function

### Question

Let $$\Omega = \mathbb{C} \setminus \{0,1\}$$ and let $$f: \Omega \to \mathbb{C}$$ be defined by $f(z) = \frac{1}{z(z-1)}.$ Show that $$f$$ has no primitive on $$\Omega$$, but that it has a primitive on $$\mathbb{C} \setminus [0,1]$$ and determine its expression.

### Answer

We have $f(z) = -\frac{1}{z} + \frac{1}{z-1}.$ The function $$z\mapsto -1/z$$ has no primitive on $$D(0,1)\setminus \{0\}$$: indeed if $$\gamma(t) = 1/2 \times e^{i2\pi t}$$, we have $\int_{\gamma} \frac{dz}{z} = i2\pi \neq 0.$ On the other hand, on the same set, $$z \mapsto \log (z-1)$$ is a primitive of $$z \mapsto 1/(z-1)$$. Hence $$f(z)$$ has no primitive.

The function $g(z) = \log \frac{z-1}{z} = \log \left(1 - \frac{1}{z} \right)$ is defined on $$\mathbb{C} \setminus [0,1]$$ and is a primitive of $$f$$. Indeed $$g(z)$$ is defined as long as neither of the conditions $$z = 0$$ and $$1 - 1/z \in \mathbb{R}_-$$ are met; they are equivalent to the condition $$z \in [0,1]$$, which is excluded. Moreover, $$g$$ satisfies $g'(z) = \frac{1/z^2}{1 - 1/z} = \frac{1}{z(z-1)}$ hence it is a primitive of $$f$$.

## Reparametrization of Paths

### Questions

Let $$\alpha:[0,1] \to \mathbb{C}$$ be a continuously differentiable path. Let $$\phi:[0,1] \to [0,1]$$ be a continuously differentiable function such that $$\phi(0) = 0$$, $$\phi(1) = 1$$ and $$\phi'(t)>0$$ for any $$t \in [0,1]$$.
1. Show that $$\beta = \alpha \circ \phi$$ is a rectifiable path which has the same initial point, terminal point and image as $$\alpha$$.

2. Prove that for any continuous function $$f: \alpha([0,1]) \to \mathbb{C}$$, $\int_{\alpha} f(z) \, dz = \int_{\beta} f(z) \, dz.$

3. Prove that the paths $$\alpha$$ and $$\beta$$ have the same length.

### Answers

1. The statement about the initial and terminal points is obvious. The one relative to the image holds because, under the assumptions that were made, the function $$\phi$$ is a bijection from $$[0,1]$$ on itself (and its inverse is also continuously differentiable).

2. We have $\int_{\beta} f(z) \, dz = \int_{0}^1 (f \circ \beta)(t) \beta'(t)\,dt = \int_{0}^1 (f \circ \alpha)(\phi(t)) \alpha'(\phi(t))\,(\phi'(t) dt).$ The change of variable $$s = \phi(t)$$ leads to $\int_{\beta} f(z) \, dz = \int_{0}^1 (f \circ \alpha)(s) \alpha'(s)\,ds = \int_{\alpha} f(z) \, dz.$

3. We have $\int_0^1 |\beta'(t)|\, dt = \int_{0}^1 |\alpha' (\phi(t)) \phi'(t)| \, dt = \int_{0}^1 |\alpha' (\phi(t))| \, \phi'(t) dt$ The change of variable $$s = \phi(t)$$ leads to $\int_0^1 |\beta'(t)|\, dt = \int_0^1 |\alpha'(s)| \, ds,$ hence the lengths of $$\alpha$$ and $$\beta$$ are equal.

## The Logarithm: Alternate Choices

### Question

Show that for any $$\alpha \in \mathbb{R}$$, the function $$z \in \mathbb{C}_{\alpha} \mapsto 1/z$$ defined on $\mathbb{C}_{\alpha} = \mathbb{C} \setminus \{r e^{i\alpha} \;|\; r \geq 0 \}.$ has a primitive; describe the set of all its primitives.

### Answer

Let $$\gamma$$ be a closed rectifiable path of $$\mathbb{C}_{\alpha}$$. The path $$\mu: [0,1] \mapsto e^{i(\pi-\alpha)} \gamma(t)$$ is closed, rectifiable and its image is included in $$\mathbb{C} \setminus \mathbb{R}_-$$. Additionally $\int_{\gamma} \frac{dz}{z} = \int_{\gamma} \frac{d(e^{i(\pi-\alpha)}z)}{e^{i(\pi-\alpha)}z} = \int_{\mu} \frac{dz}{z}.$ Since the principal value of the logarithm is a primitive if $$z \mapsto 1/z$$ on $$\mathbb{C} \setminus \mathbb{R}_-$$, the integral of $$z\mapsto 1/z$$ on $$\mu$$ is equal to zero. Therefore, there are primitives of $$z\mapsto 1/z$$ on $$\mathbb{C}_{\alpha}$$; since $$\mathbb{C}_{\alpha}$$ is connected, they all differ from an arbitrary constant.

Alternatively, we can build explicitly such a primitive: the function $f: z \mapsto \log (z e^{i(\pi - \alpha)});$ it is defined and holomorphic on $$\mathbb{C}_{\alpha}$$ and for any $$z \in \mathbb{C}_{\alpha}$$, $f'(z) = \frac{1}{z e^{i(\pi - \alpha)}} \times e^{i(\pi - \alpha)}= \frac{1}{z}.$