Complex Analysis Topic: Singularities MA201 Mathematics III Department of Mathematics IIT Guwahati August 2015 Complex Analysis Topic: Singularities 1 / 15
Zeroes of Analytic Functions A point z 0 C is a zero of order (or multiplicity) m (m 1) of f if f(z 0 ) = f (z 0 ) = = f (m 1) (z 0 ) = 0, and f (m) (z 0 ) 0. A zero of order 1 is called a simple zero. Examples. For f(z) = z(z 1) 5 : z 0 = 0 is a simple zero and z 0 = 1 is a zero of order 5. z 0 = nπ, n Z are simple zeroes of f(z) = sin(z). f(z) = (z 2 πz) sin(z) has zeroes of order two at z 0 = 0 and π. f(z) = e z has no zeroes in C. Every polynomial of order n 1 has zeroes in C with their total multiplicity n. Complex Analysis Topic: Singularities 2 / 15
If z 0 is a zero of a polynomial P (z) of multiplicity m, then P (z) = (z z 0 ) m Q(z) where Q(z) is a polynomial. In general, we have the following: Theorem Suppose f is analytic in a domain D, and z 0 D. Then z 0 is a zero of order m for f if and only if f(z) = (z z 0 ) m g(z) for some function g which is analytic in a neighborhood of z 0 and g(z 0 ) 0. Proof. ) Follows from the Taylor series ( ) f (n) (z 0 ) f(z) = (z z 0 ) n. n! n=0 ) g(z) = n=0 b n(z z 0 ) n, where b 0 = g(z 0 ) 0. Then, f(z) = b 0 (z z 0 ) m + b 0 (z z 0 ) m + must be the Taylor series of f(z). So, f(z 0 ) = f (z 0 ) = = f (m 1) (z 0 ) = 0 and f (m) (z 0 ) = (m!)b 0 0. Complex Analysis Topic: Singularities 3 / 15
Theorem For a function f, analytic on a domain D, the following are equivalent. 1 There is z 0 D such that f (n) (z 0 ) = 0 for n 0. 2 f 0 in D. 3 The set S = {z C : f(z) = 0} has a limit point in D. Proof. (1) Taylor series about z 0 is the zero series f(z 0 ) 0 in an open disc centered at z 0. given z D keep shifting the disc towards z and get f(z) = 0 (2) (2) (3) Trivial. (3) (1) Suppose z 0 is a limit point of S. We claim that f (n) (z 0 ) = 0 for all n 0. Suppose not. Let m be least such that f (m) (z 0 ) 0. Then f(z) = (z z 0 ) m g(z) where g is analytic and g(z 0 ) 0. There is r > 0 such that g(z) 0 in B r (z 0 ), because g is continuous. Thus, f(z) 0 for 0 < z z 0 < r, a contradiction. Complex Analysis Topic: Singularities 4 / 15
Corollaries For a non-constant function f on a domain D A zero of infinite order is not possible. Set of zeroes of f cannot have a limit point in D, i.e., each zero of f is isolated. If two functions f and g, analytic in a domain D, are such that the set {z D : f(z) = g(z)} has a limit point, then f(z) = g(z) for all z D. Complex Analysis Topic: Singularities 5 / 15
A point z 0 C is a singular point or a singularity of a function f, if 1 f is not analytic at z 0, and 2 every neighborhood of z 0 contains a point at which f is analytic. Example. f(z) = 1 z has a singular point z 0 = 0. f(z) = cot(z) has singular points z = nπ, n Z. Points on the negative real axis are singular points for f(z) = Log (z). A singular point z 0 C is an isolated singular point of f, if there is a deleted neighborhood 0 < z z 0 < r of z 0 throughout which f is analytic. Otherwise, z 0 is called a non-isolated singular point. Example. z 0 = 0 is an isolated singular point for 1 z, e1/z, sin(z). z z 0 = nπ are isolated singular points for cot(z). 0 and the points on the negative real axis are non-isolated singular points for Log (z). Complex Analysis Topic: Singularities 6 / 15
Classification of isolated singular points Let z 0 be a singular point of f. Then 1 z 0 is called a removable singular point, if lim z z0 f(z) exists and equals a complex number a 0. Example: z 0 = 0 is a removable singular point of sin z and ez 1. z z 2 z 0 is called a pole, if lim f(z) =. z z0 Example: z 0 = 0 is a pole for 1 z, 1 and cot z. z2 3 z 0 is called an essential singular point, lim f(z) does not exist. z z0 ( ) 1 Example: z 0 = 0 is an essential singular point of e 1/z and sin. z Complex Analysis Topic: Singularities 7 / 15
Order of a pole Note: A point z 0 is a pole of f if and only if z 0 is a zero of 1/f. Definition A pole z 0 of f is said to be of order m 1 if z 0 is a zero of order m of the function 1/f. A pole of order 1 is called a simple pole. Examples: cot z has simple poles at the points z 0 = nπ, n Z. 1 z 0 = 1 is a pole of order 2 for the function f(z) = (z 1) 2. Complex Analysis Topic: Singularities 8 / 15
Singular points and Laurent series Theorem Suppose f has an isolated singular point at z 0. Let f(z) = n= a n (z z 0 ) n be the Laurent series of f about z 0, 0 z z 0 < r for r > 0. Then 1 z 0 is a removable singularity if and only if a n = 0 for all n N. 2 z 0 is a pole of order m if and only if a m 0 and a n = 0 for all n > m. 3 z 0 is an essential singularity if and only if a n 0 for infinitely many n N. Note: If f has a removable singularity at z 0, then defining f(z 0 ) = lim z z0 f(z) = a 0, f becomes analytic at z 0. Complex Analysis Topic: Singularities 9 / 15
Examples f(z) = ez 1 = 1 + z z 2! + z2 3! + z3 + for 0 < z <. 4! So, z 0 = 0 is a removable singularity of f. Note: Defining f(0) = 1 you get f analytic at z 0 = 0. 1 f(z) = (z 1)(z 2) = 1 z 1 (z 1) n for 0 < z 1 < 1. n=0 So, z 0 = 1 is a simple pole of f. Similarly, z 0 = 2 is a simple pole of f. f(z) = z 2 exp ( ) 1 z = z 2 + z + 1 2! + ( ) 1 1 n for z > 0. (n + 2)! z n=1 So, z 0 = 0 is an essential singularity of f. Complex Analysis Topic: Singularities 10 / 15
Properties of Removable Singularity Theorem Suppose that f is analytic in 0 < z z 0 < r for some r > 0, and z 0 is a singular point of f. Then the following are equivalent: 1 z 0 is a removable singularity of f. 2 f(z) = a n(z z 0) n for 0 < z z 0 < r. n=0 3 There is an analytic function g in z z 0 < r such that g(z) = f(z) for 0 < z z 0 < r. 4 lim f(z) exists and is finite. z z 0 ( ) 5 lim (z z z z 0)f(z) = 0. 0 6 f is bounded in a deleted neighborhood of z 0. Look at the Laurent series f(z) = a n (z z + n=1 0) n n=0 If (5) or (6) holds, then a n must be zero for all n N. a n(z z 0) n for 0 < z z 0 < r. Complex Analysis Topic: Singularities 11 / 15
Properties of a Pole Theorem Suppose that f is analytic in 0 < z z 0 < r for some r > 0. Then the following are equivalent: 1 f has a pole of order m at z 0. 2 1 f 3 f(z) = 4 f(z) = has a zero of order m at z0. k= m a k (z z 0) k for 0 < z z 0 < r and a m 0. g(z), where g is analytic at z0 and g(z0) 0. (z z 0) m 5 The function (z z 0) m f(z) has a removable singularity at z 0. 6 lim z z0 f(z) =. 7 lim z z 0 (z z 0) k f(z) = 0, if k > m, a m, if k = m,, if k < m. Complex Analysis Topic: Singularities 12 / 15
Properties of Essential Singularities Theorem Suppose that f is analytic in 0 < z z 0 < r for some r > 0. Then the following are equivalent: 1 f has essential singularity at z 0, i.e., lim does not exist. z z 0 2 f(z) = a k (z z 0) k for 0 < z z 0 < r and a n 0 for infinitely many n N. k= 3 f is neither bounded nor approaches as z z 0. Theorem Picard s Great Theorem: Suppose has an essential singularity at z 0. Then in each deleted neighborhood N(z 0) of z 0, f assumes every complex number, with possibly one exception, an infinite number of times, i.e., the image of 0 < z z 0 < r under f is an infinite number of copies of C \ {atmost one point}. Complex Analysis Topic: Singularities 13 / 15
Singularity at Point at Infinity ( ) Definition Suppose that f is analytic in R < z < for some R 0. Define g by g(z) = f ( ) 1 z for 0 < z < 1/R. Then, f is said to have a removable singularity / pole of order m / essential singularity at if g has a removable singularity / pole of order m / essential singularity at 0, respectively. Examples. (1) If P (z) is a non-constant polynomial, then the function f given by f(z) = 1 has a removable singularity at z =. P (z) (2) Any non-constant polynomial P (z) has a pole at z =. (3) e z, sin(z), cos(z), sinh(z) have an essential singularity at z =. Complex Analysis Topic: Singularities 14 / 15
Result An entire function has removable singularity at if and only if it is a constant function. [Hint. For only if part, show that f is bounded.] Result An entire function has a pole of order m at if and only if it is a polynomial of degree m. Exercise. Analyze the nature of the point z = for the rational function R(z) = P (z), where P (z) and Q(z) are polynomials of Q(z) degrees m and n, respectively. Complex Analysis Topic: Singularities 15 / 15