Re: Trouble with 5.10 and what I am supposed to do.

I think my solution is correct and depends totally on removing ambiguities, since

is multivalued. Here is a more detailed explanation:

The general power Let’s define Log w=log |w| + i * arg w

This means that Log w is an infinitely many-valued function of w.

The

principal value of Log w, obtained by giving arg w its principal value, will be denoted by log w, since it is identical with the ordinary logarithm when w is real and positive.

The principal value of arg z is defined as that which satisfies –pi<arg z<=pi

If z & w denote any complex numbers then we can define the

principal value of the power to be the number

uniquely determined by the equation

= exp{z * logw}, see Penrose 5.4 (w not = 0)

where log w is the

principal value of Log w.

if we choose other values of Log w we obtain other sets of values of the power which can be called

subsidiary values.

Once we have chosen a value for Log w then

has a unique value for each z, called its principal value.

The following formula contains ALL possible values of

=exp{z * (log w + 2*k*pi*i)}. Formula 1

The principal value of

is obtained by setting k=0.

Let’s look at the case where w=e and z=1 i.e.

=

Substituting in formula 1 gives

=exp{(log e+ 2*k*pi*i)}.

We obtain the principal value for

by setting k=0

=e as expected.

Now if we look at the case where w=e and z=1+2*pi*i and apply formula 1 it gives us

= exp{(1+2*pi*i)(log e + 2*k*pi*i)}

We want to produce the principal value of

again for consistency, so we set k=0, which gives

=exp{1+2*pi*i}

These 2 values for

are

distinct principal values when z has the two values 1 and 1+2*pi*i.

It is therefore

incorrect to equate them as is done at the beginning of exercise 05.10.

This is what leads to the paradox.In order to be consistent and give

a unique value for each z then we must always use the same value of Log w. In the above I chose the principal value of Log e which is 1. This is the usual convention.