3.1 DIFFERENTIABLE AND ANALYTIC FUNCTIONS 103 functions Q(z) are Q(2)+0. #0. The then fis at points refore the we z-axis. extensions (a) Show that P' (2) a1+2a2z+...+nan"-1. (b) Show that, for k = 0,1,...,n, ak p) (0), where p() denotes the kth derivative of P. (By convention, P(0) (2) P (2) for all z.) 6. Let P be a polynomial of degree 2, given by P(2)=(2-1) (222), where 21 22. Show that P' (2) 1 1 = + cat zo. P(z) 2-21 2-22 P'O Note: The quotient is known as the logarithmic derivative of P. 7. Use L'Hôpital's rule to find the following limits. alim some V 2-i-1- 1+1 2+2 (c) lim (d) lim 2-146-22+2 -64 (e) -+1+i√ *3+8 (f) lim 23-8 --1+i√3 A 8. Use Equation (3-1) to show that 9. Show that = "=-nz"-1, where n is a positive integer. 10. Verify the identity. d dz (2) 9 (2)h(z) = f' (2) g (2) h (2) + f (2) 9' (2) h (2) + f (z)9 (2) h' (2). 11. Show that the function f (z) = |2|2 is differentiable only at the point zo = 0. Hint: To show that is not differentiable at zo 0, choose horizontal and vertical lines. through the point zo and show that A approaches two distinct values as Az→0 along those two lines. 12. Verify (a) Identity (3-4). (b) Identity (3.7) x²-4x²+ 4x2 (0+2) 20. x²+2X. -3x²+4x2 x²+ 2x X2(-3+4) x(x+2) 3.1 DIFFERENTIABLE AND ANALYTIC FUNCTIONS 101 Theorem 3.1 If f is differentiable at zo, then f is continuous at zo. Proof From Equation (3-1), we obtain f(z)-ƒ(20) lim 2430 = f'(20). *120 Using the multiplicative property of limits given by Formula (2-19), we get lim f (2) f (zo)] = lim f(z)- f (20) (2-20) 2140 2-20 2-20 f(z)-f(20) = lim lim (z-zo) 2-20 2-20 2-20 = f'(zo)-0=0. This result implies that lim f (z) = f (zo), which is equivalent to showing 2110 that is continuous at zo. We can establish Equation (3-8) from Theorem 3.1. Letting h(z) = f (2) g (2) and using Definition 3.1, we write h' (20) = lim h(z) - h (20) = lim f(z)9 (2)-f(20) 9 (20) 2120 2-20 2120 2-20 If we subtract and add the term f (zo) g (z) in the numerator, we get h' (20) = lim = 2-20 lim 2-20 f(2)g(2)-f(20) 9 (2) + f (zo) 9 (2)f (20) 9 (20) 2-20 f(2)g(2)-f(20) 9 (2) 2-20 + lim 2-20 f(zo) 9 (2)-f (20) 9 (20) 2-20 (z)-9(20)