Saturday, March 23, 2013

Branched Complex Functions (1)

It often comes as a shock or surprise to many college math students that complex functions can exhibit branches. One of the best ways of exploring this is by using the logarithmic function, i.e. based on the simple exponential equation:

e (w)  = z

Here z is any non-zero complex number.

In a more general form for a function: w = u + iv, we can rewrite our simple equation as:

e u   e v   = r exp(i q) 

From this we can see:

e u   = r   and:   e v   =  Q + 2 n p

where n is any integer.

Now, since  e u   = r is the same as u = ln r (check the properties of exponential and logs!) it can be seen the first simple exponential equation is satisfied only if w has one of the following values:

w= ln r + i(Q+ 2 n p)    with n = 0, + 1, + 2 ….

So if we then write:

log z =   ln r + i(Q+ 2 n p)     (n = 0, + 1, + 2 ….)

we obtain the simple form:

e log z   = z  or exp(log z) = z

Then, given z = r exp(i Q) 

The definition of a multi-valued function enters with the definition of log z above, in terms of ln r, iQ etc.

Then define an angle variable q which can have any one of the values:

q =  Q+ 2 n p (n = 0, + 1, + 2 ….)

Where Q = Arg z

Then the equation for log z can be recast as:

log z= ln r + i(q) 

In more detail:

log z= ln êr ê + i arg z  (z ¹ 0)

It needs to be emphasized here that the left hand side of e log z   = z   (with the order of the functions reversed) is not always equal to z.

Lastly, the principal value of log z is the value obtained when n= 0 in the earlier quation for log z (e.g. log z= ln r + i(q) ) and is always denoted Log z:

Log z = ln r + iQ

Or, alternatively:

Log z = ln êz ê + i Arg z  (z ¹ 0)

Note also that:  log z = Log z + 2 n pi      (n = 0, + 1, + 2 ….)

From the preceding it can easily be seen that Log z is well defined (and single valued) provided z ¹ 0.

Problem:  Show that Log z is well defined and single-valued when z ¹ 0.  (Hint: Use z = r exp (i0) )

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