# 2.3 Further exercises

Here are some further exercises to end this section.

## Exercise 19

Calculate the partial derivatives and of each of the following functions.

### Answer

1. Differentiating with respect to while keeping fixed, we obtain

   Differentiating with respect to while keeping fixed, we obtain

2. Here , so

3. Here , so

## Exercise 20

Calculate the partial derivatives and of each of the following functions, and evaluate these partial derivatives at .

### Answer

1. Here , so

   So, at the partial derivatives have the values

2. Here , so

   So, at the partial derivatives have the values

## Exercise 21

Find the gradient of the graph of at the point in the -direction and in the -direction.

### Answer

Since , it follows that

The gradient of the graph at in the -direction is

The gradient of the graph at in the -direction is

## Exercise 22

Use the Cauchy–Riemann equations to show that there is no point of at which the function

is differentiable.

### Answer

Writing in the form

we obtain

Hence

If is differentiable at , then the Cauchy–Riemann equations require that

that is,

But is never zero, so , which is impossible. It follows that there is no point of at which is differentiable.

## Exercise 23

Use the Cauchy–Riemann equations to show that the function

is entire, and find its derivative.

### Answer

In this case,

so

These partial derivatives are defined and continuous on the whole of . Furthermore,

so the Cauchy–Riemann equations are satisfied at every point of .

By the Cauchy–Riemann Converse Theorem, is entire, and

(So , and in fact .)

## Exercise 24

Use the Cauchy–Riemann equations to find all the points at which the following functions are differentiable, and calculate their derivatives.

### Answer

1. Here

   so

   The Cauchy–Riemann equations are satisfied only if . So cannot be differentiable at unless . Since the partial derivatives above exist, and are continuous on (and in particular when ), it follows from the Cauchy–Riemann Converse Theorem that is differentiable on the set .

   On this set,

2. Here

   so

   The Cauchy–Riemann equations are not satisfied unless and . So is not differentiable except possibly at 0. Since the partial derivatives above exist, and are continuous at , it follows from the Cauchy–Riemann Converse Theorem that is differentiable at 0. Furthermore,

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