- #1

freddyfish

- 57

- 0

**A complex series need not be defined for all z within the "circle of convergence"?**

The (complex) radius of convergence represents the radius of the circle (centered at the center of the series) in which a complex series converges.

Also, a theorem states that a (termwise) differentiated series has the same radius of convergence as the original series.

Now, Ln z is obviously singular (at least) at the negative real axis which is a distance 1 away from the z

_{0}=-1 + i. But the Taylor series of Ln z centered at z

_{0}=-1 + i has a radius of convergence equal to 2

^{0.5}. Thus, the derivative of Ln z is not defined on negative real axis, but according to the theorem it has radius of convergence R=2

^{0.5}.

This implies that a series need not be defined everywhere a distance less than R from the center of the series.

However, the definition of convergence of a complex series is that that the limit of the partial sums converge to some finite value.

How can this contradiction be eliminated?

My own thoughts about this is that this contradiction would not rise from the above definitions and theorems if the derivative of Ln z does not equal the termwise differentiated complex series representation of Ln z. If this is the case, then why?

Thanks

//Freddy