The complex logarithm is in fact the
natural logarithm of a complex number
and I know that some of you are used to
having the notation for the natural
logarithm to be log but I'm going to
stick with this one, ln since that is
one I'm used to, so what is the natural
logarithm of a complex number we know
that a complex number can be written
in this form here it has some real part
and some imaginary part and if we're
going to graph it will look something
like this right, but this is one way of
representing a complex number this
complex number right now is presented in
a rectangular form but we can also
represent a complex number by its length
the absolute value of the complex number and some
angle which are going to call alpha
and with some trigonometry we can write
the complex number on the following form
and if we factor out the length of the
complex number and use Euler's formula
we get the complex number written in its
polar form
and if we use this form instead for our
complex number and use some of the laws
for a logarithm, the natural logarithm of
a complex number becomes the following
and now we can use that the angle is the
argument for the complex number and here
it is important to remember that this
argument can be many different values
since there are infinite number of
angles that corresponds to the same
complex number, I mean if you have angle
and your add or subtract 2pi you
will still end up on the exact same spot
in the complex plane and that is why we
need to define a new kind of argument
for complex numbers, this new argument is
called the principal argument and is
denoted the same way as before but this
time with a capital A at the start
and its formal definition is that the
principal argument is the angle for the
complex number that lies between minus
pi and pi and can therefore only have
one value so in short the principal
argument is just one number just one
angle for the complex number while the
argument of a complex number is all
possible angles and the relationship
between these two is that the argument
for a complex number is equal to the
principal argument plus n times 2 pi for
some n and by using this new knowledge
the whole formula for the natural
logarithm of a complex number becomes
the following
this formula is also called the
multivalued logarithm, since it takes on
many different values for the same
complex number but sometimes we are just
looking for one value the principal
value which is denoted by Ln with a
capital L and this principle value of
the logarithm is when n is equal to zero
in the multivariable formula
and one important thing to remember
about these two formulas is that the
only work if the complex number is not
equal to zero since the natural logarithm
is not defined for 0
lets continue by doing an example, here I
want us to determine the multivalued
logarithm of the complex number 1+i
and also its principal value
to solve this problem we need to know two things
we need to know the absolute value of the
complex number and we need to know its
principal argument and we can determine
these two from this figure
by using the Pythagorean theorem we know
that the absolute value of this complex
number is equal to the square root of
2 and by using the arc tangent
function we know that the principal
argument is equal to pi/4
and now the only thing that's left is to
plug in the values into the formulas so
the multivalued logarithm will become
the following
and the principal value is used the
multivalued with n is equal to zero
so that one becomes the following
thanks for watching!
