Exercise 21.  Explain how the fundamental theorem of calculus studied in complex analysis and
the fundamental theorem of calculus studied in calculus are different.  
How are they similar?  

Solution 21.

See text and/or instructor's solution manual.

Answer.  In ordinary calculus the Fundamental Theorem of calculus applies only to functions of a real variable that are integrated over a closed interval  .

In complex analysis the Fundamental Theorem of calculus applies only to analytic functions in a simply connected domain D, where    and    are two points in D joined by a contour C.  

The methods appear similar because analytic functions are often times the complex extensions of real functions.   

Hence the rules for computing anti-derivatives appear to be similar for analytic functions,

and the formula    looks similar to the one introduced in ordinary calculus.  

This solution is complements of the authors.

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