MATH 141 Chapter Notes - Chapter 4: Mathematical Induction, Formula 4, Riemann Sum
Document Summary
Cauchy"s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting and useful properties of analytic functions. If you learn just one theorem this week it should be cauchy"s integral formula! We start with a statement of the theorem for functions. After some examples, we"ll give a gener- alization to all derivatives of a function. After some more examples we will prove the theorems. After that we will see some remarkable consequences that follow fairly directly from the cauchy"s formula. Theorem 4. 1. (cauchy"s integral formula) suppose is a simple closed curve and the function () is analytic on a region containing and its interior. Cauchy"s integral formula: simple closed curve , () analytic on and inside . This is remarkable: it says that knowing the values of on the boundary curve means we know everything about inside !!
