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1 2-1 (z - 1)3 (b) lim = 8; (c) lim z² + 1 z→∞z - 1 = C

1 2-1 (z - 1)3 (b) lim = 8; (c) lim z² + 1 z→∞z - 1 = C

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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10. Use the theorem in Sec. 17 to show that 4z² (a) lim z→∞ (z-1)² = 4: 1 (b) lim z→1 (z − 1)³ and = ∞; z² + 1 z→∞ z - 1 (c) lim ∞.
Transcribed Image Text:10. Use the theorem in Sec. 17 to show that 4z² (a) lim z→∞ (z-1)² = 4: 1 (b) lim z→1 (z − 1)³ and = ∞; z² + 1 z→∞ z - 1 (c) lim ∞.
50 ANALYTIC FUNCTIONS (1) 17. LIMITS INVOLVING THE POINT AT INFINITY It is sometimes convenient to include with the complex plane the point at infinity, denoted by ∞o, and to use limits involving it. The complex plane together with this point is called the extended complex plane. To visualize the point at infinity, one can think of the complex plane as passing through the equator of a unit sphere centered at the origin (Fig. 25). To each point z in the plane there corresponds exactly one point P on the surface of the sphere. The point P is the point where the line through z and the north pole N intersects the sphere. In like manner, to each point P on the surface of the sphere, other than the north pole N, there corresponds exactly one point z in the plane. By letting the point N of the sphere correspond to the point at infinity, we obtain a one to one correspondence between the points on the sphere and the points in the extended complex plane. The sphere is known as the Riemann sphere, and the correspondence is called a stereographic projection. N 1 lim f(z) = wo Z-ZO FIGURE 25 CHAP. 2 gibre Observe that the exterior of the unit circle centered at the origin in the complex plane corresponds to the upper hemisphere with the equator and the point N deleted. Moreover, for each small positive number &, those points in the complex plane exterior set |z| > 1/8 a neighborhood of ∞o. to the circle |z| = 1/8 correspond to points on the sphere close to N. We thus call the Let us agree that in referring to a point z, we mean a point in the finite plane. mentioned. Hereafter, when the point at infinity is to be considered, it will be specifically A meaning is now readily given to the statement lim f(z) = ∞o if Z-Zo when either zo or wo, or possibly each of these numbers, is replaced by the point at infinity. In the definition of limit in Sec. 15, we simply replace the appropriate theorem illustrates how this is done. neighborhoods of zo and wo by neighborhoods of oo. The proof of the following lovigen = 0 Theorem. If zo and wo are points in the z and w planes, respectively, then 1 lim z→zo f(z)
Transcribed Image Text:50 ANALYTIC FUNCTIONS (1) 17. LIMITS INVOLVING THE POINT AT INFINITY It is sometimes convenient to include with the complex plane the point at infinity, denoted by ∞o, and to use limits involving it. The complex plane together with this point is called the extended complex plane. To visualize the point at infinity, one can think of the complex plane as passing through the equator of a unit sphere centered at the origin (Fig. 25). To each point z in the plane there corresponds exactly one point P on the surface of the sphere. The point P is the point where the line through z and the north pole N intersects the sphere. In like manner, to each point P on the surface of the sphere, other than the north pole N, there corresponds exactly one point z in the plane. By letting the point N of the sphere correspond to the point at infinity, we obtain a one to one correspondence between the points on the sphere and the points in the extended complex plane. The sphere is known as the Riemann sphere, and the correspondence is called a stereographic projection. N 1 lim f(z) = wo Z-ZO FIGURE 25 CHAP. 2 gibre Observe that the exterior of the unit circle centered at the origin in the complex plane corresponds to the upper hemisphere with the equator and the point N deleted. Moreover, for each small positive number &, those points in the complex plane exterior set |z| > 1/8 a neighborhood of ∞o. to the circle |z| = 1/8 correspond to points on the sphere close to N. We thus call the Let us agree that in referring to a point z, we mean a point in the finite plane. mentioned. Hereafter, when the point at infinity is to be considered, it will be specifically A meaning is now readily given to the statement lim f(z) = ∞o if Z-Zo when either zo or wo, or possibly each of these numbers, is replaced by the point at infinity. In the definition of limit in Sec. 15, we simply replace the appropriate theorem illustrates how this is done. neighborhoods of zo and wo by neighborhoods of oo. The proof of the following lovigen = 0 Theorem. If zo and wo are points in the z and w planes, respectively, then 1 lim z→zo f(z)
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