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
Differentiating
with respect to 
while keeping 
fixed, we obtain 
Differentiating with respect to
while keeping fixed, we obtain 
Here
, so 
Here
, so 
Exercise 20
Calculate the partial derivatives and of each of the following functions, and evaluate these partial derivatives at 
.
Answer
Here
, so 
So, at
the partial derivatives have the values 
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
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,
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, 
OpenLearn - Introduction to complex analysis
Except for third party materials and otherwise, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence, full copyright detail can be found in the acknowledgements section. Please see full copyright statement for details.