Exercise 13.  Explain how the limit definition for the derivative in complex analysis and the limit definition for the derivative in calculus are different.  How are they similar?  

Solution 13.

See text and/or instructor's solution manual.

Solution.  The form of the definition is identical, but the meaning is more subtle in the complex case.  

For starters, the limit must exist when    from any direction in the complex case.  

The real case is limited to two directions.

Furthermore, when     there are "infinitely many" ways to approach    and the limit must always end up with the same value.

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell