Ch.4:Complex Integration Ch.4:Complex lntegration L4.1 Contours L4.2 Contour Integrals LThe Length of a Contour The Length of a Contour(Cont'd) Introduction (Cont'd) (is a geometric quantity that depends only on the point We will accomplish this by first defining the integral along a set y and is independent of the particular admissible single directed smooth curve and then defining integrals along parametrization used in the computation a contour in terms of the integrals along its smooth components The length of a contour is simply defined to be the sum of the length of its component curves Finally,we once again obtain simple rules for evaluating integrals in terms of antiderivatives +口·0+t。年之，220C 4日10。+之+1生，意0G Ch.4:Complex Integration Ch.4:Commplex lntegration L4.2 Contour Integrals L4.2 Contour Integrals Introduction Riemann Sum In calculus,the definite integral of a real-valued function f Ca over an interval [a,b]is defined as the limit of certain sums ∑k=1f(ck)△rk(called Riemann sums) However,the fundamental theorem of calculus lets us evaluate ”23 integrals more directly when an antiderivative is known The aim of this section is to use this notion of Riemann sums to define integral of a complex-valued function along a contour T in the z-plane Partitioned Curve +口·811定+1意1意00Ch.4: Complex Integration 4.1 Contours The Length of a Contour The Length of a Contour (Cont’d) l(γ) is a geometric quantity that depends only on the point set γ and is independent of the particular admissible parametrization used in the computation The length of a contour is simply defined to be the sum of the length of its component curves Ch.4: Complex Integration 4.2 Contour Integrals Introduction In calculus, the definite integral of a real-valued function f  over an interval [a, b] is defined as the limit of certain sums n k=1 f(ck)xk (called Riemann sums) However, the fundamental theorem of calculus lets us evaluate integrals more directly when an antiderivative is known The aim of this section is to use this notion of Riemann sums to define integral of a complex-valued function along a contour Γ in the z-plane Ch.4: Complex Integration 4.2 Contour Integrals Introduction (Cont’d) We will accomplish this by first defining the integral along a single directed smooth curve and then defining integrals along a contour in terms of the integrals along its smooth components Finally, we once again obtain simple rules for evaluating integrals in terms of antiderivatives Ch.4: Complex Integration 4.2 Contour Integrals Riemann Sum