Morera’s theorem12 9. Cauchy’s formula We indicate the proof of the following, as we did in class. Theorem 1.1 (Cauchy). The starting point is the following. Theorem 4.5. Let f(z) be an analytic function de ned on a simply connected re-gion Denclosed by a piecewise smooth curve Cgoing once around counterclockwise. Simultaneously, we expect a relation to complex di erentiation, extending the fundamental theorem of single-variable calculus: when f= F0for complex-di erentiable F on open set , it should be that, for any path from z 1 to z 2 inside , Z the converse of Cauchy’s theorem. In the next few lectures we will explore this theme, and prove theorems that will form the basis of all that we will accomplish in the rest of the course. 3.Among its consequences is, for example, the Fundamental Theorem of Algebra, which says that every nonconstant complex polynomial has at least one complex zero. Considering Theorem 2, all we need to show is that Z f(z)dz= 0 LECTURE 7: CAUCHY’S THEOREM The analogue of the fundamental theorem of calculus proved in the last lecture says in particular that if a continuous function f has an antiderivative F in a domain D; then R C f(z)dz = 0 for any given closed contour lying entirely on D: Now, two questions arises: 1) Under what conditions on f we can guarantee the Teaching the Fundamental Theorem of Calculus: A Historical Reflection - Integration from Cavalieri to Darboux ... Cauchy's definition of continuity [4, Section 2.2, p. 26] would seem to correspond to our definition of uniform continuity, especially if we take at face value his statement in terms of infinitesimals. Then I C f (z)dz = 0 whenever C is a simple closed curve in R. It is trivialto show that the traditionalversion follows from the basic version of the Cauchy Theorem. It is somewhat remarkable, that in many situations the converse also holds true. Then Z f(z)dz= 0 for all closed paths contained in U. I’ll prove it in a somewhat informal way. Proof. That is, we have a formula for all the derivatives, so in particular the derivatives all exist. 2.The result itself is known as Cauchy’s Integral Theorem. The Cauchy-Taylor theorem11 8. The Cauchy integral formula10 7. Theorem 3 (Morera’s theorem). THE CAUCHY INTEGRAL FORMULA AND THE FUNDAMENTAL THEOREM OF ALGEBRA D. ARAPURA 1. Contour integration1 2. Proof. Cauchy’s theorem for homotopic loops7 5. Local integrability6 4. the fundamental theorem of calculus. Then, \(f\) has derivatives of all order. Theorem 15.4 (Traditional Cauchy Integral Theorem) Assume f isasingle-valued,analyticfunctiononasimply-connectedregionRinthecomplex plane. Goursat’s theorem5 3. So, now we give it for all derivatives f(n)(z) of f. This will include the formula for functions as a special case. Evaluation of real de nite integrals8 6. THE FUNDAMENTAL THEOREMS OF FUNCTION THEORY TSOGTGEREL GANTUMUR Contents 1. Theorem \(\PageIndex{1}\) Suppose \(f(z)\) is analytic on a region \(A\). By fundamental theorem of calculus, the assumption of the theorem implies that f(z) has a primitive F(z). Cauchy’s integral formula is worth repeating several times. If f(z) is continuous in open UˆC and satisfies Z f(z)dz= 0 for any closed loop ˆU, then f(z) is holomorphic. Cauchy’s Theorem Cauchy’s theorem actually analogue of the second statement of the fundamental theorem of calculus and integration of familiar functions is facilitated by this result In this example, it is observed that is nowhere analytic and so need not be independent of choice of the curve connecting the points 0 and . Our version of the fundamental theorem of complex analysis, known as Cauchy’s Theorem can be stated as follows: Theorem 3 Assume fis holomorphic in the simply connected region U. Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science The Cauchy Integral Formula THEOREM 1. This follows from Cauchy’s integral formula for derivatives.