Singularities Lecture 16 Singularities Singularities Behavior of following functions f at 0: 1 f (z) = z 9 sin z f (z) = z ez − 1 f (z) = z 1 f (z) = 1 sin( z ) f (z) = Log z 1 f (z) = e z In the above we observe that all the functions are not analytic at 0; however in every neighborhood of 0 there is a point at which f is analytic. Lecture 16 Singularities Singularities Definition: The point z0 is called a singular point or singularity of f if f is not analytic at z0 but every neighborhood of z0 contains at least one point at which f is analytic. ez − 1 1 Here ; ; sin 1 ; Log z etc. has singularity at z = 0: z z 2 z z¯; jzj2; Re z; Im z; zRe z are nowhere analytic. That does not mean that every point of C is a singularity. A singularities are classified into TWO types: 1 A singular point z0 is said to be an isolated singularity or isolated singular point of f if f is analytic in B(z0; r) n fz0g for some r > 0. 2 A singular point z0 is said to be an non-isolated singularity if z0 is not an isolated singular point. sin z 1 ; ; sin( 1 ) (0 is isolated singular point). z z 2 z 1 ; Log z these functions has non-isolated singularity at 0. sin(π=z) Lecture 16 Singularities Laurent's Theorem Suppose that 0 ≤ r < R: Let f be an analytic defined on the annulus A = ann(a; r; R) = fz : r < jz − aj < Rg: Then for each z 2 A; f (z) has the Laurrent series representation 1 X n f (z) = an(z − a) n=−∞ where the convergence is absolute and uniform in ann(a; r1; R1) if r < r1 < R1 < R: The coefficients are given by 1 Z f (z) an = n+1 dz 2πi γ (z − a) where γ(t) = a + seit ; t 2 [0; 2π] and for any r < s < R. −1 X n Moreover, this series is unique and an(z − a) is called( principal part) k=−∞ 1 X n and an(z − a) is called( regular/analytic part). k=0 Lecture 16 Singularities Singularities If f has an isolated singularity at z0, then f is analytic in B(z0; r) n fz0g for some r > 0. In this case f has the following Laurent series expansion: a−n a−1 2 f (z) = ··· n + ··· + + a0 + a1(z − a) + a2(z − z0) + ··· : (z − z0) (z − z0) If all a−n = 0 for all n 2 N, then the point z = z0 is a removal singularity. The point z = z0 is called a pole if all but a finite number of a−n's are non-zero. If m is the highest integer such that a−m 6= 0, then z0 is a Pole of order m. 0 If a−n 6= 0 for infinitely many n s, then the point z = z0 is a essential singularity. The term a−1 is called residue of f at z0. Lecture 16 Singularities Removable singularities The following statements are equivalent: 1 f has a removable singularity at z0. 2 If all a−n = 0 for all n 2 N. 3 lim f (z) exists and finite. z!z0 4 lim (z − z0)f (z) = 0. z!z0 5 f is bounded in a deleted neighborhood of z0: sin z The function has removable singularity at 0. z ez − 1 at z = 0: 2z z 2 − 4 at z = 2: z − 2 Lecture 16 Singularities Pole The following statements are equivalent: f has a pole of order m at z0. g(z) f (z) = m , g is analytic at z0 and g(z0) 6= 0: (z − z0) 1 has a zero of order m. f lim jf (z)j = 1: z!z0 m+1 lim (z − z0) f (z) = 0 z!z0 m lim (z − z0) f (z) has removal singularity at z0: z!z0 2z at z = 3. (z − 3)2 sin z at z = 0. z 2 Lecture 16 Singularities Essential singularity The following statements are equivalent: f has a essential singularity at z0. The point z0 is neither a pole nor removable singularity. lim f (z) does not exists. z!z0 Infinitely many terms in the principal part of Laurent series expansion around the point z0. sin(1=z) at z = 0 z 3 sin(1=z) at z = 0 Lecture 16 Singularities Singularities at 1 Let f be a complex valued function. Define another function g by 1 g(z) = f : z Then the nature of singularity of f at z = 1 is defined to be the the nature of singularity of g at z = 0. f (z) = z 3 has a pole of order 3 at 1. ez has an essential singularity at 1. An entire function f has a removal singularity at 1 if and only if f is constant.(Prove This!) An entire function f has a pole of order m at 1 if and only if f is a polynomial of degree m.(Prove This!) Lecture 16 Singularities.