The fact that the integral in (4) is zero when n is an integer follows only part

Question

The fact that the integral in (4) is zero when n is an integer follows only partially from the Cauchy-Goursat theorem. When n is zero or a negative integer, 1/z- zo is a polynomial e, n =-3, l/(Z-Za)-3-(z-3) and therefore entire. Theorem i 8.2.1 then im- plies fe dziz-Zn)" = O. It is left as an exercise to show that the integral is still zero when n is a positive integer different from one. See Problem 22 in Exercises 18.2 EXAMPLE 5 Applying Formula (4) Evaluate +2-3 uate _5z + dz, where Cis the circle lz - 21-2 SOLUTION Since the denominator factors as+2z-3-(e-Ixe+ 3), the integrand fails to be analytic at z 1 and z--3. Of these two points, only z = 1 lies within the contour C, which is a circle centered at Z 2 of radius-2. Now by partial fractions, 5z + 732 12759 5z +7 and so "J c+3 In view of the result given in (4), the first Goursat theorem, the value of the second in in (5) 2mi. By the Cauchy- integral is zero. Hence, (5) becomes 5z +7 C) If C, C, and C, are the simple closed contours shown in FIGURE 1826 and if fis analytic on each of the three contours as well as at each point interior to C but exterior to both C, and C2, then by innoducing cuts, we get inn Theorem 18.2.1 thatf(Odz +/Qdz + fodk-o. Hence, FIGURE 1826 Triply connected domain D The next theorem summarizes the general result for a multiply connected domain with n·bles": Theorem 18.22 Cauchy-Goursat Theorem for Multiply Connected Domains Suppose C, C,..,C, are simple closed curves with apositive orientation such that C,Cc are interior to C but the regions interior to each C, k = 1, 2, , n, have no points in com- mon. If f is analytic on each contour and at each point interior to C but exterior to all the G,, k = 1,2, ,n,then

Explanation / Answer

|z-2| = 2

|x+iy - 2| = 2

(x-2)^2 +y^2 = 4 ( of the form (x – h)2 + (y – k)2 = r2)

which is a circle with center 2,0 and radius 2 So this is the graph

so the first point z = 1 will only lie in this circle and z = -3 wont lie inside this circle outside the circle

since z =1 is lying insider the circle we apply cauchy's residue where 1/z-a integration over curve c will be 2*pi*a *i since z =1 means a = 1 we have answer 2*pi*a and the other term will be zero as it is lying outside the curve

Related Questions

Navigate

• Browse (All)
• Browse T
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.