We know that generally

(see my post on the general power

)

Itâ€™s important to make the distinction between

1)

which is multi-valued and

2)

.... which is single valued

Looking at 1) above for z=1 we obtain

Just as (in one of my earlier posts) I used Log w to represent the multi-valued logarithm of w, and log w to represent the principal value, such that

,

so I could use E(z) to represent the multi-valued exp (z) and e the principal value:

, where k is any integer value, positive or negative.

if we take k in the range -2 to +2 then we get:

(the e we all know and love!)

These are all separate valid values for

Writing

as is done in Ex 05.10, is not a valid thing to do. It is the same as saying:

And therefore

.

A simpler, more obvious example of a similar fallacy would be for example sqrt(9). This is also multi-valued and has the 2 possible distinct values -3 and +3.

However, writing

and then saying

Therefore +3=-3 is clearly fallacious.

Or, another example

but we know

therefore

fallacious again.

When we say, for example above, that +3=-3 because they are both equal to sqrt(9), we are forgetting that sqrt(9) is a multi-valued function and therefore we are not allowed to equate the two values.

This is exactly the same as saying

is equal to

.

If we take logarithms then we can see the fallacy immediately:

and it is clear that