lecture10 - The logarithmic function Let z C Consider exp w...

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The logarithmic function. Let z C . Consider exp w = z. ( ) By above, if w 1 is a solution of (*) then so is w 1 + 2 nπi . Each of these values is called a logarithm of z and is written log z . If x R , x > 0 then exp w = x has a unique real solution. We call this log x . In (*), let w = u + iv : z = exp w = exp u (cos v + i sin v ) , so | z | = exp u and v is a value of arg z . Then u = log | z | so log z = u + iv = log | z | + i arg z - a many valued function . The principal value of log z is the value when π < arg z π . We write this Log z : Log z = log z + i arg z, π < arg z π. Definition. The complex plane with the negative real axis, including zero, removed is called the cut plane and written C π . (Picture.) The following in intuitively obvious but takes a little effort to proof rigorously. Due to time constraints we shall omit the proof. Lemma 5.3. [S&Tpages122-125] The function arg z is continuous on the cut plane. From it we deduce: Theorem 5.4. [S&Tpage126] The function Log z is continuous on the cut plane.
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