By W. Chen
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Additional resources for Introduction to Complex Analysis Lecture notes
An ∈ C are arbitrary. Show that the polynomial P (z) − P (z0 ) is divisible by z − z0 . c) Deduce that P (z) is divisible by z − z0 if P (z0 ) = 0. d) Suppose that a polynomial P (z) of degree n vanishes at n distinct values z1 , z2 , . . , zn ∈ C, so that P (z1 ) = P (z2 ) = . . = P (zn ) = 0. Show that P (z) = c(z − z1 )(z − z2 ) . . (z − zn ), where c ∈ C is a constant. e) Suppose that a polynomial P (z) of degree n vanishes at more than n distinct values. Show that P (z) = 0 identically.
Suppose also that C2 is the semicircle from 1 to −1 through −i, followed in the negative (clockwise) direction. Show that z 3 dz = C1 z 3 dz and z dz = C2 C1 z dz.
A function f is said to be analytic at a point z0 ∈ C if it is diﬀerentiable at every z in some -neighbourhood of the point z0 . The function f is said to be analytic in a region if it is analytic at every point in the region. The function f is said to be entire if it is analytic in C. 1. Consider the function f (z) = |z|2 . In our usual notation, we clearly have u = x2 + y 2 and v = 0. The Cauchy-Riemann equations 2x = 0 and 2y = 0 can only be satisﬁed at z = 0. It follows that the function is diﬀerentiable only at the point z = 0, and is therefore analytic nowhere.