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Lecture 11 29Th Aug Lecture 11 : Isolated Sungularities: continued Application to Evaluation of Real Integrals Trigonometric Integrals Improper Integrals INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012 Anant R. Shastri August 29, 2012 Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Application to Evaluation of Real Integrals Trigonometric Integrals Improper Integrals Lecture 11 : Isolated Sungularities: continued Essential Singularities Singularity at infinity Residues Application to Evaluation of Real Integrals Trigonometric Integrals Improper Integrals Anant R. Shastri IITB MA205 Complex Analysis I A natural question is what happens when we allow infinitely many terms. I Of course, we need to assume that such an ‘infinite sum' is convergent Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity I The singular part of a function around a pole consisted finitely many negative powers of z − a: Anant R. Shastri IITB MA205 Complex Analysis I Of course, we need to assume that such an ‘infinite sum' is convergent Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity I The singular part of a function around a pole consisted finitely many negative powers of z − a: I A natural question is what happens when we allow infinitely many terms. Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity I The singular part of a function around a pole consisted finitely many negative powers of z − a: I A natural question is what happens when we allow infinitely many terms. I Of course, we need to assume that such an ‘infinite sum' is convergent Anant R. Shastri IITB MA205 Complex Analysis I What kind of singularity f has at a? I Answer depends on how many terms bm are non zero. Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity P m I Let m≥1 bmt be a power series of infinite radius of convergence. Then for any a 2 C the P −m sum f (z) = m bm(z − a) makes sense for all z 6= a and defines a holomorphic function f in C n fag: Anant R. Shastri IITB MA205 Complex Analysis I Answer depends on how many terms bm are non zero. Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity P m I Let m≥1 bmt be a power series of infinite radius of convergence. Then for any a 2 C the P −m sum f (z) = m bm(z − a) makes sense for all z 6= a and defines a holomorphic function f in C n fag: I What kind of singularity f has at a? Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity P m I Let m≥1 bmt be a power series of infinite radius of convergence. Then for any a 2 C the P −m sum f (z) = m bm(z − a) makes sense for all z 6= a and defines a holomorphic function f in C n fag: I What kind of singularity f has at a? I Answer depends on how many terms bm are non zero. Anant R. Shastri IITB MA205 Complex Analysis I It is a pole iff only finitely many and at least one of bm are non zero. I It is the third type that we are interested in, now. Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity I The point a is a removable singularity iff all bm = 0: Anant R. Shastri IITB MA205 Complex Analysis I It is the third type that we are interested in, now. Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity I The point a is a removable singularity iff all bm = 0: I It is a pole iff only finitely many and at least one of bm are non zero. Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity I The point a is a removable singularity iff all bm = 0: I It is a pole iff only finitely many and at least one of bm are non zero. I It is the third type that we are interested in, now. Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity Let z = a be an isolated singularity of f 6≡ 0. It may turn out that z = a is neither a removable singularity i.e., limz!a(z − a)f (z) 6= 0, nor a pole, i.e., limz−a f (z) 6= 1. Such a singularity is called an essential singularity. Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity It can be proved that if f has an essential singularity at a; then there is a holomorphic function g in a nbd P m of a and a power series m≥1 bmt of infinite radius convergence with infinitely many bm 6= 0 such that X −m f (z) = bm(z − a) + g(z) m≥1 where g is holomorphic in a nbd of a: Anant R. Shastri IITB MA205 Complex Analysis I We have lim f (x) = 1; lim f (x) = 0 x!0+ x!0− and jf ({y)j = 1 for all y 2 R: Thus lim z ! 0f (z) does not exist nor we have limz!0 jf (z)j = 1: I Therefore z = 0 is an essential singularity of e1=z : Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity: Example 1=z 1 1 I f (z) = e = 1 + + + ··· ; z 6= 0: z 2!z2 Anant R. Shastri IITB MA205 Complex Analysis I Therefore z = 0 is an essential singularity of e1=z : Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity: Example 1=z 1 1 I f (z) = e = 1 + + + ··· ; z 6= 0: z 2!z2 I We have lim f (x) = 1; lim f (x) = 0 x!0+ x!0− and jf ({y)j = 1 for all y 2 R: Thus lim z ! 0f (z) does not exist nor we have limz!0 jf (z)j = 1: Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Essential Singularity: Example 1=z 1 1 I f (z) = e = 1 + + + ··· ; z 6= 0: z 2!z2 I We have lim f (x) = 1; lim f (x) = 0 x!0+ x!0− and jf ({y)j = 1 for all y 2 R: Thus lim z ! 0f (z) does not exist nor we have limz!0 jf (z)j = 1: I Therefore z = 0 is an essential singularity of e1=z : Anant R. Shastri IITB MA205 Complex Analysis I To begin with we need that the function is defined and holomorphic in a neighborhood of infinity, i.e., in jzj > M for some sufficiently large M: Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Singularity at Infinity I The discussion of isolated singularity can be carried out for the point z = 1 as well. Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Singularity at Infinity I The discussion of isolated singularity can be carried out for the point z = 1 as well. I To begin with we need that the function is defined and holomorphic in a neighborhood of infinity, i.e., in jzj > M for some sufficiently large M: Anant R. Shastri IITB MA205 Complex Analysis I (1) is the same as saying lim jw nf (1=w)j = 0: (2) w!0 Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Singularity at Infinity I We say that 1 is a removable singularity or a pole of f iff f (z) lim = 0 (1) z!1 zn for some integer n: (If this integer can be chosen to be ≤ 1; then 1 is a removable singularity, otherwise, it is a pole.) Anant R. Shastri IITB MA205 Complex Analysis Lecture 11 : Isolated Sungularities: continued Essential Singularities Application to Evaluation of Real Integrals Singularity at infinity Trigonometric Integrals Residues Improper Integrals Singularity at Infinity I We say that 1 is a removable singularity or a pole of f iff f (z) lim = 0 (1) z!1 zn for some integer n: (If this integer can be chosen to be ≤ 1; then 1 is a removable singularity, otherwise, it is a pole.) I (1) is the same as saying lim jw nf (1=w)j = 0: (2) w!0 Anant R.
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