Lecture 10 24Th Aug

Lecture 10 Radius of Convergence Singularities

INDIAN INSTITUTE OF TECHNOLOGY BOMBAY MA205 Complex Analysis Autumn 2012

Anant R. Shastri

August 24, 2012

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities

Lecture 10

Radius of Convergence

Singularities Removable Singularities Poles

Anant R. Shastri IITB MA205 Complex Analysis I Consider sn = sup{bn, bn+1,...}. Then sn is a monotonically decreasing sequence. So, the following makes sense.

I Deﬁnition: Limsupn{bn} := lim sn. n→∞

Lecture 10 Radius of Convergence Singularities Recall Limsup

I Given a sequence of real numbers {bn}, let us recall what we mean by lim supn{bn}.

Anant R. Shastri IITB MA205 Complex Analysis I Deﬁnition: Limsupn{bn} := lim sn. n→∞

Lecture 10 Radius of Convergence Singularities Recall Limsup

I Given a sequence of real numbers {bn}, let us recall what we mean by lim supn{bn}.

I Consider sn = sup{bn, bn+1,...}. Then sn is a monotonically decreasing sequence. So, the following makes sense.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Recall Limsup

I Given a sequence of real numbers {bn}, let us recall what we mean by lim supn{bn}.

I Consider sn = sup{bn, bn+1,...}. Then sn is a monotonically decreasing sequence. So, the following makes sense.

I Deﬁnition: Limsupn{bn} := lim sn. n→∞

Anant R. Shastri IITB MA205 Complex Analysis I Two important properties of the Limsup are:

I (Limsup-I) If α > lim supn{bn} then there exists n0 such that for all n ≥ n0 we have, bn < α.

I (Limsup-II) If β < lim supn{bn} then there

exists inﬁnitely many nj such that bnj > β.

Lecture 10 Radius of Convergence Singularities Recall Limsup

I This is also the same as the least upper bound of the set of limits of all convergent subsequences of {bn}.

Anant R. Shastri IITB MA205 Complex Analysis I (Limsup-I) If α > lim supn{bn} then there exists n0 such that for all n ≥ n0 we have, bn < α.

I (Limsup-II) If β < lim supn{bn} then there

exists inﬁnitely many nj such that bnj > β.

Lecture 10 Radius of Convergence Singularities Recall Limsup

I This is also the same as the least upper bound of the set of limits of all convergent subsequences of {bn}.

I Two important properties of the Limsup are:

Anant R. Shastri IITB MA205 Complex Analysis I (Limsup-II) If β < lim supn{bn} then there

exists inﬁnitely many nj such that bnj > β.

Lecture 10 Radius of Convergence Singularities Recall Limsup

I This is also the same as the least upper bound of the set of limits of all convergent subsequences of {bn}.

I Two important properties of the Limsup are:

I (Limsup-I) If α > lim supn{bn} then there exists n0 such that for all n ≥ n0 we have, bn < α.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Recall Limsup

I This is also the same as the least upper bound of the set of limits of all convergent subsequences of {bn}.

I Two important properties of the Limsup are:

I (Limsup-I) If α > lim supn{bn} then there exists n0 such that for all n ≥ n0 we have, bn < α.

I (Limsup-II) If β < lim supn{bn} then there

exists inﬁnitely many nj such that bnj > β.

Anant R. Shastri IITB MA205 Complex Analysis I It is important to note that lim sup always exists and can be any value in [−∞, ∞]. When a sequence is convergent (including ±∞) the

lim supn of the sequence will be equal to the limit.

Lecture 10 Radius of Convergence Singularities Recall Limsup

I Indeed, lim sup can be characterized by these two properties.

Anant R. Shastri IITB MA205 Complex Analysis I Indeed, lim sup can be characterized by these two properties.

Lecture 10 Radius of Convergence Singularities Recall Limsup

I It is important to note that lim sup always exists and can be any value in [−∞, ∞]. When a sequence is convergent (including ±∞) the

lim supn of the sequence will be equal to the limit.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard Theorem

Deﬁnition P n A formal power series P(t) = n ant is said to be convergent if there exists a non zero number z (real or complex) such that the series of complex P n numbers n anz is convergent.

Anant R. Shastri IITB MA205 Complex Analysis Convention: 1 1 = ∞; = 0. 0 ∞

Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard Theorem

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard Theorem

Theorem P n (Cauchy-Hadamard) Let P = n≥0 ant be a power series over C. Then (a) there exists 0 ≤ R ≤ ∞ such that for all 0 < r < R, the series P(z) is absolutely and uniformly convergent in |z| ≤ r and for all |z| > R the series is divergent. 1 (b) = lim sup pn |a |, R n Convention: 1 1 = ∞; = 0. 0 ∞ Anant R. Shastri IITB MA205 Complex Analysis I So, let 0 < r < R. Choose r < s < R. Then 1/s > 1/R and hence by property (Limsup-I), we must have n0 such that for all n ≥ n0, pn |an| < 1/s.

I Therefore, for all |z| ≤ r, n n |anz | < (r/s) , n ≥ n0.

Lecture 10 Radius of Convergence Singularities Radius of Convergence:

I Proof: Indeed, all that we have to prove is to take R as given by (b) and show that it satisﬁes (a).

Anant R. Shastri IITB MA205 Complex Analysis I Therefore, for all |z| ≤ r, n n |anz | < (r/s) , n ≥ n0.

Lecture 10 Radius of Convergence Singularities Radius of Convergence:

I Proof: Indeed, all that we have to prove is to take R as given by (b) and show that it satisﬁes (a).

I So, let 0 < r < R. Choose r < s < R. Then 1/s > 1/R and hence by property (Limsup-I), we must have n0 such that for all n ≥ n0, pn |an| < 1/s.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Radius of Convergence:

I Proof: Indeed, all that we have to prove is to take R as given by (b) and show that it satisﬁes (a).

I So, let 0 < r < R. Choose r < s < R. Then 1/s > 1/R and hence by property (Limsup-I), we must have n0 such that for all n ≥ n0, pn |an| < 1/s.

I Therefore, for all |z| ≤ r, n n |anz | < (r/s) , n ≥ n0.

Anant R. Shastri IITB MA205 Complex Analysis I Thus we have proved that inside the disc |z| < R the series is convergent.

Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard

I Since r/s < 1, by Weierstrass majorant criterion, it follows that P(z) is absolutely and uniformly convergent.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard

I Since r/s < 1, by Weierstrass majorant criterion, it follows that P(z) is absolutely and uniformly convergent.

I Thus we have proved that inside the disc |z| < R the series is convergent.

Anant R. Shastri IITB MA205 Complex Analysis I Then 1/s < 1/R, and hence by (Limsup-II) there exist inﬁnitely many nj , for which pnj |anj | > 1/s. nj nj I This means that |anj z | > (R/s) . It follows n that the sequence {anz } is unbounded and P n hence the series n anz is divergent. ♠

Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard Theorem:

I On the other hand, suppose |z| > R. We ﬁx s such that |z| > s > R.

Anant R. Shastri IITB MA205 Complex Analysis nj nj I This means that |anj z | > (R/s) . It follows n that the sequence {anz } is unbounded and P n hence the series n anz is divergent. ♠

Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard Theorem:

I On the other hand, suppose |z| > R. We ﬁx s such that |z| > s > R.

I Then 1/s < 1/R, and hence by (Limsup-II) there exist inﬁnitely many nj , for which pnj |anj | > 1/s.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Cauchy-Hadamard Theorem:

I On the other hand, suppose |z| > R. We ﬁx s such that |z| > s > R.

I Then 1/s < 1/R, and hence by (Limsup-II) there exist inﬁnitely many nj , for which pnj |anj | > 1/s. nj nj I This means that |anj z | > (R/s) . It follows n that the sequence {anz } is unbounded and P n hence the series n anz is divergent. ♠

Anant R. Shastri IITB MA205 Complex Analysis I The second part of the theorem gives you the formula for it. This is called the Cauchy-Hadamard formula.

I Thus the collection of all points at which a given power series converges consists of an open disc centered at the origin and perhaps some points on the boundary of the disc.

Lecture 10 Radius of Convergence Singularities Radius of Convergence:

Deﬁnition The number R obtained in the above theorem is called the radius of convergence of P(t).

Anant R. Shastri IITB MA205 Complex Analysis I Thus the collection of all points at which a given power series converges consists of an open disc centered at the origin and perhaps some points on the boundary of the disc.

Lecture 10 Radius of Convergence Singularities Radius of Convergence:

Deﬁnition The number R obtained in the above theorem is called the radius of convergence of P(t).

I The second part of the theorem gives you the formula for it. This is called the Cauchy-Hadamard formula.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Radius of Convergence:

Deﬁnition The number R obtained in the above theorem is called the radius of convergence of P(t).

I The second part of the theorem gives you the formula for it. This is called the Cauchy-Hadamard formula.

I Thus the collection of all points at which a given power series converges consists of an open disc centered at the origin and perhaps some points on the boundary of the disc.

Anant R. Shastri IITB MA205 Complex Analysis I Also observe that the theorem does not say anything about the convergence of the series at points on the boundary |z| = R.

Lecture 10 Radius of Convergence Singularities Radius of Convergence:

I Observe that if P(z) is convergent for some z, then from the last part of (a), the radius of convergence of P is at least |z|.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Radius of Convergence:

I Observe that if P(z) is convergent for some z, then from the last part of (a), the radius of convergence of P is at least |z|.

I Also observe that the theorem does not say anything about the convergence of the series at points on the boundary |z| = R.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Radius of Convergence:

Theorem Any given power series, its derived series and its integrated series all have the same radius of convergence.

Anant R. Shastri IITB MA205 Complex Analysis We have α α(α − 1) ··· (α − n + 1) a = = . n n n!

Lecture 10 Radius of Convergence Singularities Newton’s Binomial Series

Example Consider the principal branch f (z) of (1 + z)α in the open disc |z| < 1. The MacLaurin’s series P∞ n f (z) = n=0 anz is called the Newton’s binomial series.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Newton’s Binomial Series

Example Consider the principal branch f (z) of (1 + z)α in the open disc |z| < 1. The MacLaurin’s series P∞ n f (z) = n=0 anz is called the Newton’s binomial series. We have α α(α − 1) ··· (α − n + 1) a = = . n n n!

Anant R. Shastri IITB MA205 Complex Analysis If r > 1 then (1 + z)α is complex diﬀerentiable as many times as you like at z = −1. which leads to a contradiction.

Lecture 10 Radius of Convergence Singularities Newton’s Binomial Series continued:

Since f (z) is holomorphic in |z| < 1 we know that the radius of convergence r is at least 1.

Anant R. Shastri IITB MA205 Complex Analysis which leads to a contradiction.

Lecture 10 Radius of Convergence Singularities Newton’s Binomial Series continued:

Since f (z) is holomorphic in |z| < 1 we know that the radius of convergence r is at least 1. If r > 1 then (1 + z)α is complex diﬀerentiable as many times as you like at z = −1.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Radius of Convergence Singularities Newton’s Binomial Series continued:

Since f (z) is holomorphic in |z| < 1 we know that the radius of convergence r is at least 1. If r > 1 then (1 + z)α is complex diﬀerentiable as many times as you like at z = −1. which leads to a contradiction.

Anant R. Shastri IITB MA205 Complex Analysis I Deﬁnition A point z ∈ U is called an isolated singularity of f if f is deﬁned and holomorphic in a neighborhood of z except perhaps at z.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Singularity

I Deﬁnition Let U be a region in C. If f (z) is a function on a subset of U then the points at which f is not deﬁned or those points at which f is not holomorphic are referred to as singularities.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Singularity

I Deﬁnition A point z ∈ U is called an isolated singularity of f if f is deﬁned and holomorphic in a neighborhood of z except perhaps at z.

Anant R. Shastri IITB MA205 Complex Analysis I (ii) Since for any holomorphic function f , the zeros of f are isolated, it follows that all the singularities of 1/f are isolated.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Examples

I (i) If p(z) is a polynomial, then 1/p(z) has all its singularities isolated and these are nothing but the zeros of p(z).

Anant R. Shastri IITB MA205 Complex Analysis I (iii) Natural examples of holomorphic functions which have non isolated singularities are branches of logarithmic function and inverse-trigonometric functions. For instance, Ln(z) has singularities along the negative real axis.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Examples

I (i) If p(z) is a polynomial, then 1/p(z) has all its singularities isolated and these are nothing but the zeros of p(z).

I (ii) Since for any holomorphic function f , the zeros of f are isolated, it follows that all the singularities of 1/f are isolated.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Examples

I (i) If p(z) is a polynomial, then 1/p(z) has all its singularities isolated and these are nothing but the zeros of p(z).

I (ii) Since for any holomorphic function f , the zeros of f are isolated, it follows that all the singularities of 1/f are isolated.

I (iii) Natural examples of holomorphic functions which have non isolated singularities are branches of logarithmic function and inverse-trigonometric functions. For instance, Ln(z) has singularities along the negative real axis. Anant R. Shastri IITB MA205 Complex Analysis I Later on we weakened this hypothesis to include those functions f which are holomorphic on U1 = U \ A, where A is a ﬁnite subset and

I satisfying the weaker condition that f is continuous at a ∈ A.

I (Or just bounded in nbd of a.)

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I In obtaining Cauchy’s theorem and integral formulae, in the previous chapter, we began with the hypothesis that the function under consideration is holomorphic throughout a domain U.

Anant R. Shastri IITB MA205 Complex Analysis I satisfying the weaker condition that f is continuous at a ∈ A.

I (Or just bounded in nbd of a.)

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I In obtaining Cauchy’s theorem and integral formulae, in the previous chapter, we began with the hypothesis that the function under consideration is holomorphic throughout a domain U.

I Later on we weakened this hypothesis to include those functions f which are holomorphic on U1 = U \ A, where A is a ﬁnite subset and

Anant R. Shastri IITB MA205 Complex Analysis I (Or just bounded in nbd of a.)

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I In obtaining Cauchy’s theorem and integral formulae, in the previous chapter, we began with the hypothesis that the function under consideration is holomorphic throughout a domain U.

I Later on we weakened this hypothesis to include those functions f which are holomorphic on U1 = U \ A, where A is a ﬁnite subset and

I satisfying the weaker condition that f is continuous at a ∈ A.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I In obtaining Cauchy’s theorem and integral formulae, in the previous chapter, we began with the hypothesis that the function under consideration is holomorphic throughout a domain U.

I Later on we weakened this hypothesis to include those functions f which are holomorphic on U1 = U \ A, where A is a ﬁnite subset and

I satisfying the weaker condition that f is continuous at a ∈ A.

I (Or just bounded in nbd of a.)

Anant R. Shastri IITB MA205 Complex Analysis I On the other hand, diﬀerentiation under the integral sign tells us that the function deﬁned by the integral on the RHS of (1) is holomorphic in the interior of the disc. All that we need is the function f (z) is continuous on C = ∂D.

I Therefore, it follows that f is holomorphic even at points of A.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I Thus 1 Z f (z)dz f (w) = (1) 2πı C z − w is valid for all w ∈ int D \ A.

Anant R. Shastri IITB MA205 Complex Analysis I Therefore, it follows that f is holomorphic even at points of A.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I Thus 1 Z f (z)dz f (w) = (1) 2πı C z − w is valid for all w ∈ int D \ A.

I On the other hand, diﬀerentiation under the integral sign tells us that the function deﬁned by the integral on the RHS of (1) is holomorphic in the interior of the disc. All that we need is the function f (z) is continuous on C = ∂D.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

I Thus 1 Z f (z)dz f (w) = (1) 2πı C z − w is valid for all w ∈ int D \ A.

I Therefore, it follows that f is holomorphic even at points of A. Anant R. Shastri IITB MA205 Complex Analysis Definition An isolated singularity z0 of a holomorphic function is called a removable singularity if f (z0) can be defined in such a way that f becomes complex differentiable at z0, i.e., there exists a holomorphic function g : U → C such that for all z ∈ U \{a} we have f (z) = g(z).

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

Let us consolidate this simple observation.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

Let us consolidate this simple observation. Definition An isolated singularity z0 of a holomorphic function is called a removable singularity if f (z0) can be defined in such a way that f becomes complex differentiable at z0, i.e., there exists a holomorphic function g : U → C such that for all z ∈ U \{a} we have f (z) = g(z).

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

Theorem Let U be a domain, a ∈ U be any point. Suppose f is holomorphic in U \{a}. A necessary and suﬃcient condition that there exists a holomorphic function g : U → C such that g(z) = f (z), z ∈ U \{a} is that f is continuous at a.

Anant R. Shastri IITB MA205 Complex Analysis I Conversely, suppose the above limit exists. Take a circular region D around a contained in U and so that f is holomorphic in D \{a}. It is enough to ﬁnd a holomorphic function g on D such that f (z) = g(z) for all z ∈ D \{a}.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularity

I Proof: If such a g exists as stated in the deﬁnition, then

lim f (z) = lim g(z) = g(a) z→a z→a exists.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularity

I Proof: If such a g exists as stated in the deﬁnition, then

lim f (z) = lim g(z) = g(a) z→a z→a exists.

I Conversely, suppose the above limit exists. Take a circular region D around a contained in U and so that f is holomorphic in D \{a}. It is enough to ﬁnd a holomorphic function g on D such that f (z) = g(z) for all z ∈ D \{a}.

Anant R. Shastri IITB MA205 Complex Analysis I On the other hand, if we take g(z) as RHS, then we know that g is holomorphic function throughout D. (by diﬀerentiating under the integral sign). This completes the proof. ♠

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularity

I But then C.I.F. says that 1 Z f (w) f (z) = dw 2πı ∂D w − z for all z ∈ D and not equal to a.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularity

I But then C.I.F. says that 1 Z f (w) f (z) = dw 2πı ∂D w − z for all z ∈ D and not equal to a.

I On the other hand, if we take g(z) as RHS, then we know that g is holomorphic function throughout D. (by diﬀerentiating under the integral sign). This completes the proof. ♠

Anant R. Shastri IITB MA205 Complex Analysis I Obviously z = 0 is an isolated singularity. We easily see that lim f (z) exists. Hence, z = 0 is a z→0 removable singularity. Also we see that lim f (z) = 1. So we can deﬁne f (0) = 1 and z→0 make f holomorphic at z = 0 also. ez −1 I Other similar example are z , z cot z etc. for which z = 0 is a removable singularity.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities: Examples

sin z I Consider the function f (z) = , z 6= 0. z

Anant R. Shastri IITB MA205 Complex Analysis ez −1 I Other similar example are z , z cot z etc. for which z = 0 is a removable singularity.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities: Examples

sin z I Consider the function f (z) = , z 6= 0. z I Obviously z = 0 is an isolated singularity. We easily see that lim f (z) exists. Hence, z = 0 is a z→0 removable singularity. Also we see that lim f (z) = 1. So we can deﬁne f (0) = 1 and z→0 make f holomorphic at z = 0 also.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities: Examples

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Removable Singularities

Remark One easy way a removable singularity z0 can arise is by taking a genuine holomorphic function f around this point and then brutally redeﬁning the value of f to be something else only at z0 or merely pretending as if f is not deﬁned at z0.

Anant R. Shastri IITB MA205 Complex Analysis I Theorem Let z = a be a pole of f (z) deﬁned and holomorphic in U \{a}. Then there exists a positive integer k k such that in a disc around a, limz→a(z − a) f (z) k+1 exists; (equivalently limz→a(z − a) f (z) = 0).

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Deﬁnition Let now z = a be an isolated singularity of f . We say a is a pole of f if

lim |f (z)| = ∞. z→a

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Deﬁnition Let now z = a be an isolated singularity of f . We say a is a pole of f if

lim |f (z)| = ∞. z→a

I Theorem Let z = a be a pole of f (z) deﬁned and holomorphic in U \{a}. Then there exists a positive integer k k such that in a disc around a, limz→a(z − a) f (z) k+1 exists; (equivalently limz→a(z − a) f (z) = 0). Anant R. Shastri IITB MA205 Complex Analysis I Consider g(z) = 1/f (z) on Bδ(a) \{a} =: U1. Then g(z) is holomorphic on U1. Moreover, limz→a g(z) = 0.

I Hence z = a is a removable singularity of g(z). We are forced to deﬁne g(a) = 0 in order to obtain a holomorphic function on Bδ(a).

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Proof: Since limz→a |f (z)| = ∞ implies that there is δ > 0 such that f (z) 6= 0 in Bδ(a).

Anant R. Shastri IITB MA205 Complex Analysis I Hence z = a is a removable singularity of g(z). We are forced to deﬁne g(a) = 0 in order to obtain a holomorphic function on Bδ(a).

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Proof: Since limz→a |f (z)| = ∞ implies that there is δ > 0 such that f (z) 6= 0 in Bδ(a).

I Consider g(z) = 1/f (z) on Bδ(a) \{a} =: U1. Then g(z) is holomorphic on U1. Moreover, limz→a g(z) = 0.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Proof: Since limz→a |f (z)| = ∞ implies that there is δ > 0 such that f (z) 6= 0 in Bδ(a).

I Consider g(z) = 1/f (z) on Bδ(a) \{a} =: U1. Then g(z) is holomorphic on U1. Moreover, limz→a g(z) = 0.

I Hence z = a is a removable singularity of g(z). We are forced to deﬁne g(a) = 0 in order to obtain a holomorphic function on Bδ(a).

Anant R. Shastri IITB MA205 Complex Analysis I A partial converse is also true. k+1 I For, it follows that the function (z − a) f (z) is holomorphic around a and vanishes at a.

I If the order of zero at a is m then we have (z − a)k+1f (z) = (z − a)mα(z) for a holomorphic function α with α(a) 6= 0.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Suppose a is a zero of g of order k. Then g(z) = (z − a)k h(z) for a holomorphic function h on U1 and h(a) 6= 0. Therefore, k limz→a(z − a) f (z) = limz→a 1/h(z) exists. This k+1 implies limz→a(z − a) f (z) = 0.

Anant R. Shastri IITB MA205 Complex Analysis k+1 I For, it follows that the function (z − a) f (z) is holomorphic around a and vanishes at a.

I If the order of zero at a is m then we have (z − a)k+1f (z) = (z − a)mα(z) for a holomorphic function α with α(a) 6= 0.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I A partial converse is also true.

Anant R. Shastri IITB MA205 Complex Analysis I If the order of zero at a is m then we have (z − a)k+1f (z) = (z − a)mα(z) for a holomorphic function α with α(a) 6= 0.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I A partial converse is also true. k+1 I For, it follows that the function (z − a) f (z) is holomorphic around a and vanishes at a.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I A partial converse is also true. k+1 I For, it follows that the function (z − a) f (z) is holomorphic around a and vanishes at a.

I If the order of zero at a is m then we have (z − a)k+1f (z) = (z − a)mα(z) for a holomorphic function α with α(a) 6= 0.

Anant R. Shastri IITB MA205 Complex Analysis I If m < k + 1 then

α(z) |f (z)| = → ∞ (z − a)k−m+1 as z → a.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I If m were bigger than or equal to k + 1 then it follows that f (z) = (z − a)m−k−1α(z) and so z = a is a removable singularity.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I If m were bigger than or equal to k + 1 then it follows that f (z) = (z − a)m−k−1α(z) and so z = a is a removable singularity.

I If m < k + 1 then

α(z) |f (z)| = → ∞ (z − a)k−m+1 as z → a.

Anant R. Shastri IITB MA205 Complex Analysis I Deﬁnition If the order is 1 then the pole is called a simple pole; if the order is bigger than 1, then the pole is called a multiple pole.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Deﬁnition The least such integer k is called the order of the pole of f at z = a.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Deﬁnition The least such integer k is called the order of the pole of f at z = a.

I Deﬁnition If the order is 1 then the pole is called a simple pole; if the order is bigger than 1, then the pole is called a multiple pole.

Anant R. Shastri IITB MA205 Complex Analysis I Deﬁnition A function which has all its singularities, if any, as poles, is called a meromorphic function in U. [Observe that the poles of a meromorphic function are required to be isolated.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Indeed we have just proved that f (z) = (z − a)−k h(z), for all z in a neighborhood of a, where h(z) is holomorphic and h(a) 6= 0, where k is the order of the pole. The number k with this property is unique.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles

I Deﬁnition A function which has all its singularities, if any, as poles, is called a meromorphic function in U. [Observe that the poles of a meromorphic function are required to be isolated.

Anant R. Shastri IITB MA205 Complex Analysis P(z) I (ii) More generally, functions of the form Q(z) where P, Q are polynomials are meromorphic functions.

I (iii) Sums, products and scalar multiples of meromorphic functions are meromorphic.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Examples

I (i) The simplest example of a function with a pole at z = 0 of order k is 1/zk .

Anant R. Shastri IITB MA205 Complex Analysis I (iii) Sums, products and scalar multiples of meromorphic functions are meromorphic.

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Examples

I (i) The simplest example of a function with a pole at z = 0 of order k is 1/zk . P(z) I (ii) More generally, functions of the form Q(z) where P, Q are polynomials are meromorphic functions.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Examples

I (i) The simplest example of a function with a pole at z = 0 of order k is 1/zk . P(z) I (ii) More generally, functions of the form Q(z) where P, Q are polynomials are meromorphic functions.

I (iii) Sums, products and scalar multiples of meromorphic functions are meromorphic.

Anant R. Shastri IITB MA205 Complex Analysis I (v) A typical example of this type is (sin z)/z, which is indeed a holomorphic function.

I (vi) Amongst trigonometric functions we have tan z and cot z which have inﬁnitely many poles each of order 1. This follows easily since the order of the zeros of sin z are all 1. Therefore same is true of cos z also,(since cos z = sin(z + π/2.)

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Examples

I (iv) If f and g are non zero meromorphic functions then so is f /g. Further, the zeros of g become poles of f /g, in general. However, if z = a is a common zero of f and g, it becomes a removable singularity of f /g provided the order of the zero of f at a is bigger than or equal to that of g.

Anant R. Shastri IITB MA205 Complex Analysis I (vi) Amongst trigonometric functions we have tan z and cot z which have inﬁnitely many poles each of order 1. This follows easily since the order of the zeros of sin z are all 1. Therefore same is true of cos z also,(since cos z = sin(z + π/2.)

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Examples

I (v) A typical example of this type is (sin z)/z, which is indeed a holomorphic function.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Examples

I (v) A typical example of this type is (sin z)/z, which is indeed a holomorphic function.

I (vi) Amongst trigonometric functions we have tan z and cot z which have inﬁnitely many poles each of orderAnant 1. R.This Shastri followsIITB MA205 easily Complex Analysis since the order of the zeros of sin z are all 1. Therefore same is true of cos z also,(since cos z = sin(z + π/2.) I For z 6= a, we can divide this expression by (z − a)k and write 0 0 0 b1 = bk−1, b2 = bk−2,..., bk = b0, to obtain b b b f (z) = k + k−1 +···+ 1 +φ(z) (z − a)k (z − a)k−1 (z − a)

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Singular part

I Let f have a pole of order k at z = a and consider h(z) = (z − a)k f (z), and apply the Taylor’s expansion to h(z) = 0 0 0 k−1 k b0 +b1(z −a)+···+bk−1(z −a) +φ(z)(z −a)

where φk is holomorphic at z = a.

Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Singular part

I Let f have a pole of order k at z = a and consider h(z) = (z − a)k f (z), and apply the Taylor’s expansion to h(z) = 0 0 0 k−1 k b0 +b1(z −a)+···+bk−1(z −a) +φ(z)(z −a)

where φk is holomorphic at z = a.

I For z 6= a, we can divide this expression by (z − a)k and write 0 0 0 b1 = bk−1, b2 = bk−2,..., bk = b0, to obtain b b b f (z) = k + k−1 +···+ 1 +φ(z) (z − a)k (z − a)k−1 (z − a)

Anant R. Shastri IITB MA205 Complex Analysis I Further, if we write Taylor’s expansion for φ(z) on the rhs above we get Laurent1 expansion for f (z).

Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Singular part

I The sum of terms which involve bi is called the principal part of f (z) at z = a. Observe that f minus its principal part is a holomorphic function.

1Pierre Alphonse Laurent (1813-1854) was a French Engineer cum mathematician who proved this theorem around 1843. Anant R. Shastri IITB MA205 Complex Analysis Lecture 10 Removable Singularities Radius of Convergence Poles Singularities Poles: Singular part

I The sum of terms which involve bi is called the principal part of f (z) at z = a. Observe that f minus its principal part is a holomorphic function.

I Further, if we write Taylor’s expansion for φ(z) on the rhs above we get Laurent1 expansion for f (z).

1Pierre Alphonse Laurent (1813-1854) was a French Engineer cum mathematician who proved this theorem around 1843. Anant R. Shastri IITB MA205 Complex Analysis