Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f"

Transcription

1 1 Bernstein's analytic continuation of complex powers c1995, Paul Garrett, garrettmath.umn.edu version January 27, 1998 Analytic continuation of distributions Statement of the theorems on analytic continuation Bernstein's proofs Proof of the Lemma: the Bernstein polynomial Proof of the Proposition: estimates on zeros

2 Garrett: `Bernstein's analytic continuation of complex powers' 2 Let f be a polynomial in x 1 ; : : : ; x n with real coecients. For complex s, let f s + be the function dened by f s +(x) = f(x) s if f(x) 0 f s +(x) = 0 if f(x) 0 Certainly for <(s) 0 the function f s + is locally integrable. For s in this range, we can dened a distribution, denoted by the same symbol f s +, by f s +() := R n f s +(x) (x) dx where is in Cc 1 (R n ), the space of compactly-supported smooth real-valued functions on R n. The object is to analytically continue the distribution f+, s as a meromorphic (distribution-valued) function of s. This type of question was considered in several provocative examples in I.M. Gelfand and G.E. Shilov's Generalized Functions, volume I. (One should also ask about analytic continuation as a tempered distribution). In a lecture at the 1963 Amsterdam Congress, I.M. Gelfand rened this question to require further that one show that the `poles' lie in a nite number of arithmetic progressions. Bernstein proved the result in 1967, under a certain `regularity' hypothesis on the zero-set of the polynomial f. (Published in Journal of Functional Analysis and Its Applications, 1968, translated from Russian). The present discussion includes some background material from complex function theory and from the theory of distributions.

3 Garrett: `Bernstein's analytic continuation of complex powers' 3 1 Analytic continuation of distributions First we recall the nature of the topologies on test functions and on distributions. Let Cc 1 (U) be the collection of compactly-supported smooth functions with support inside a set U R n. As usual, for U compact, we have a countable family of seminorms on Cc 1 (U): (f) := sup jd fj x where for = ( 1 ; : : : ; n ) we write, as usual, D = 1 n : : : x 1 x n It is elementary to show that (for U compact) Cc 1 (U) is a complete, locally convex topological space (a Frechet space). To treat U not necessarily compact (e.g., R n itself) let U 1 U 2 : : : be compact subsets of U so that their union is all of U. Then Cc 1 (U) is the union of the spaces Cc 1 (U i ), and we give it with the locally convex direct limit topology. The spaces D (U) and D (R n ) of distributions on U and on R n, respectively, are the continuous duals of D(U) = Cc 1 (U) and D(R n ) = Cc 1 (R n ). For present purposes, the topology we put on the continuous dual space V of a topological vectorspace V is the weak- topology: a sub-basis near 0 in V is given by sets U v; := f 2 V : j(v)j < g In this context, a V -valued function f on an open subset of C is holomorphic on if, for every v 2 V, the C-valued function z! f(z)(v) on is holomorphic in the usual sense. This notion of holomorphy might be more pedantically termed `weak- holomorphy', since reference to the topology might be required. If z o 2 for an open subset of C and f is a holomorphic V -valued function on? z o, say that f is weakly meromorphic at z o if, for every v 2 V, the C-valued function z! f(z)(v) has a pole (as opposed to essential singularity) at z o. Say that f is strongly meromorphic at z o if the orders of these poles are bounded independently of v. That is, f is strongly meromorphic

4 Garrett: `Bernstein's analytic continuation of complex powers' 4 at z o if there is an integer n and an open set containing z o so that, for all v 2 V the C-valued function z! (z? z o ) n f(z)(v) is holomorphic on. If n is the least integer f so that (z?z o ) n f is holomorphic at z o, then f is of order?n at z o, etc. To say that f is strongly meromorphic on an open set is to require that there be a set S of points of with no accumulation point in so that f is holomorphic on? S, and so that f is strongly meromorphic at each point of S. For brevity, but risking some confusion, we will often say `meromorphic' instead of `strongly meromorphic'. 2 Statement of the theorems on analytic continuation Let O be the polynomial ring R[x 1 ; : : : ; x n ]. For z 2 R n, let O z be the local ring at z, i.e., the ring of ratios P=Q of polynomials where the denominator does not vanish at z. Let m z be the maximal ideal of O z consisting of elements of O z whose numerator vanishes at z. Let I z (depending upon f) be the ideal in O z generated by f ; : : : ; f x 1 x n A point z 2 R n is simple with respect to the polynomial f if f(z) = 0 for some N we have I z m N z There are i 2 m z so that f = P i i f xi Remarks: The second condition is equivalent to the assertion that O z =I z is nite-dimensional. The simplest situation in which the second condition holds is when I z = O z, i.e., some partial derivative of f is non-zero at z. The third condition does not follow from the rst two. For example, Bernstein points out that with f(x; y) = x 5 + y 5 + x 2 y 2 the rst two conditions hold but the third does not. Theorem (local version): If z is a simple point with respect to f, then there is a neighborhood U of x so that the distribution f s +;U() := f s +(x) (x) dx

5 Garrett: `Bernstein's analytic continuation of complex powers' 5 on test functions 2 Cc 1 (U) on U has an analytic continuation to a meromorphic element in the continuous dual of Cc 1 (U). Theorem (global version): If all real zeros of f(x) are simple (with respect to f), then f s + is a meromorphic (distribution-valued) function of s 2 C. 3 Bernstein's proof Let R z be the ring of linear dierential operators with coecients in O z. Note that R z is both a left and a right O-module: for D 2 R z, for f; g 2 O and a smooth function near z, the denition is (fdg)() := f D(g) Lemma: There is a dierential operator D 2 R z and a non-zero `Bernstein polynomial' H in a single variable so that D(f n+1 ) = H(n)f n for any natural number n. (Proof below.) Proof of Local Theorem from Lemma: Let U be a small-enough neighborhood of z so that on it all coecients of D are holomorphic on U. For suciently large natural numbers n the function f n+1 + is continuously dierentiable, so we have For each xed 2 C 1 c Df n+1 + = H(n)f n + (U) consider the function g(s) := (Df s+1 +? H(s)f s +)() The hypotheses of the proposition below are satised, so the equality for all large-enough natural numbers implies equality everywhere: This gives us Df s+1 + = H(s)f s + f s + = Df s+1 + H(s) Now we claim that for any 0 n 2 the distribution f s + on Cc 1 (U) is meromorphic for <(s) >?n. For n = 0 this is certain. The formula just derived then gives the induction step. Further, this argument makes clear that the `poles' of f s + restricted to C1 c (U) are concentrated on the nite collection of arithmetic progressions i ; i? 1; i? 2; : : :

6 Garrett: `Bernstein's analytic continuation of complex powers' 6 where the i are the roots of H(s). In particular, the order of the pole of f s + at a point s o is equal to the number of roots i so that s o lies among i ; i? 1; i? 2; : : : In particular, the distribution f s + really is (strongly) meromorphic. This proves the Theorem, granting the Lemma and granting the Proposition. Proposition (attributed by Bernstein to `Carlson'): If g is an analytic function for <s > 0 and jg(s)j < be c<s and if g(n) = 0 for all suciently large natural numbers n, then g 0. Proof of Global Theorem: Invoking the Local Theorem and its proof above, for each z 2 R n we choose a neighborhood U z of z in which f s + is meromorphic, so that U z is ariski-open, i.e., is the complement of a nite union of zero sets of polynomials. Indeed, writing f(x) = X i f with i 2 O z, as in the proof of the Local Theorem, let i = g i =h i with polynomials g i and h i, and take U z to be the complement of the union of the zero-sets of the denominators h i. Then Hilbert's Basis Theorem implies that the whole R n is covered by nitely-many U z1 ; : : : ; U zn of these ariski-opens. Then make a partition of unity P subordinate to this nite cover, i.e., take 1 ; : : : ; n so that i 0, i 1, and spt i U zi. Then f s + = X i if s + By choice of the neighborhoods U zi, the right-hand side is a nite sum of meromorphic (distribution-valued) functions. 4 Proof of the Lemma: the Bernstein polynomial Now we prove existence of the dierential operator D and the 'Bernstein polynomial' H. This is the most serious part of this proof. (The complex function theory proposition is not entirely trivial, but is approximately standard). Proof of Lemma: Let where the i 2 m z are so that P := X i 2 R z f = X i f 2 R z

7 Garrett: `Bernstein's analytic continuation of complex powers' 7 Also put Then we have S i := f P? f = f (P + 1)? Q P (f) = X i i f = f Thus, by Leibniz' formula, S i f = f f? f f = 0 P (f n ) = nf n S i (f n ) = 0 Sublemma: There is a non-zero polynomial M in one variable so that M(P ) can be written in the form M(P ) = X i for some J i 2 R z. Proof of Sublemma: Write, as usual, For a natural number m, write J i f jj = 1 + : : : + n P m = X jjm D m; where m; 2 O z. That is, we move all the coecients to the right of the dierential operators. That this is possible is easy to see: for example, x i? x i x j x j is 0 or?1 as i = j or not. Further, the coecients m; are polynomials in the i. Thus, Then taking M(P ) of the form m; 2 m jj z M(P ) = X mq b m P m = X m; D b m m;

8 Garrett: `Bernstein's analytic continuation of complex powers' 8 with b m 2 R, the condition of the sublemma will be met if X m b m m; 2 I z for all indices. If jj N, where I m N, then this condition is automatically fullled. Thus, there are nitely-many conditions X m b m m; = 0 where m; is the image of m; in O z =m N z. Since the latter quotient is, by hypothesis, a nite-dimensional vector space, the collection of such conditions gives a nite collection of homogeneous equations in the coecients b m. More specically, there are dim O z =m N z cardf : jj < Ng such conditions. By taking q large enough we assure the existence of a nontrivial solution fb m g. This proves the Sublemma. Returning to the proof of the Lemma: as an equation in R z Now put Then we have M(P )(P + 1) = X J i (P + 1) = X J i S i + X J i X D = J i H(P ) = M(P )(P + 1) X D(f n+1 ) = ( J i f)(f n ) = = H(P )(f n ) = H(n)f n f as desired. This proves the Lemma, constructing the dierential operator D. 5 Proof of the Proposition: estimates on zeros The result we need is a standard one from complex function theory, although it is not so elementary as to be an immediate corollary of Cauchy's Theorem: Proposition: If g is an analytic function for <s > 0 and jg(s)j < be c<s and if g(n) = 0 for all suciently large natural numbers n, then g 0. Proof of Proposition: Consider G(z) := e?c g( z + 1 z? 1 )

9 Garrett: `Bernstein's analytic continuation of complex powers' 9 Then g is turned into a bounded function G on the disc, with zeros at points (n? 1)=(n + 1) for suciently large natural numbers n. We claim that, for a bounded function G on the unit disc with zeros i, either G 0 or X (1? j i j) < +1 i If we prove this, then in the situation at hand the natural numbers are mapped to n := (n? 1)=(n + 1) = 1? 1 n + 1 so here X n (1? j n j) = X n 1 n + 1 = +1 Thus, we would conclude G 0 as desired. We recall Jensen's formula: for any holomorphic function G on the unit disc with G(0) 6= 0 and with zeros 1 ; : : :, for 0 < r < 1 we have jg(0)j jijr r 1 j i j = exp 2? log jg(re i )j d Granting this, our assumed boundedness of G on the disc gives us an absolute constant C so that for all N jg(0)j jijr r j i j C (We can harmlessly divide by a suitable power of z to guarantee that G(0) 6= 0.) Then, letting r! 1, j i j jg(0)j?1 C?1 For an innite product of positive real numbers j i j less than 1 to have a value > 0, it is elementary that we must have X i (1? j i j) < +1 as claimed. This proves the proposition. While we're here, let's recall the proof of Jensen's formula (e.g., as in Rudin's Real and Complex Analysis, page 308). Fix 0 < r < 1 and let H(z) := G(z) r2? z r(? z)? z where the rst product is over roots with jj < r and the second is over roots with jj = r. Then H is holomorphic and non-zero in an open disk of radius

10 Garrett: `Bernstein's analytic continuation of complex powers' 10 r + for some > 0. Thus, log jhj is harmonic in this disk, and we have the mean value property log jh(0)j = 1 log jh(re i )j d 2? On one hand, jh(0)j = jg(0)j r jj On the other hand, if jzj = r the factors have absolute value 1. Thus, where r 2? z r(? z) log jh(re i )j = log jg(re i )j? X As noted in Rudin (see below), e i arg = jj=r log j1? e i(?arg ) j 1 2? log j1? e i j d = 0 Therefore, the integral appearing in the assertion of the mean value property is unchanged upon replacing H by G. Putting this all together gives Jensen's formula. And let's do the integral computation, following Rudin. There is a function (z) on the open unit disc so that exp((z)) = 1? z since the disc is simply-connected. We uniquely specify this by requiring that (0) = 0. We have <(z) = log j1? zj and j=(z)j < 2 Let > 0 be small. Let? =? be the path which goes (counterclockwise) around the unit circle from e i to e (2?)i and let = be the path which goes (clockwise) around a small circle centered at 1 from e (2?)i to e i. Then log j1? e i 1 j d = lim!0 2 2? log j1? e i j d =

11 Garrett: `Bernstein's analytic continuation of complex powers' 11 1 = < 2i 1 = < 2i? (z) dz = z (z) dz z by Cauchy's theorem. Elementary estimates show that the latter integral has a bound of the form which goes to 0 as! 0. C log(1=)

Topological vectorspaces

Topological vectorspaces (July 25, 2011) Topological vectorspaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Natural non-fréchet spaces Topological vector spaces Quotients and linear maps More topological

More information

Theorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r

Theorem [Mean Value Theorem for Harmonic Functions] Let u be harmonic on D(z 0, R). Then for any r (0, R), u(z 0 ) = 1 z z 0 r 2. A harmonic conjugate always exists locally: if u is a harmonic function in an open set U, then for any disk D(z 0, r) U, there is f, which is analytic in D(z 0, r) and satisfies that Re f u. Since such

More information

Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016

Department of Mathematics, University of California, Berkeley. GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 Department of Mathematics, University of California, Berkeley YOUR 1 OR 2 DIGIT EXAM NUMBER GRADUATE PRELIMINARY EXAMINATION, Part A Fall Semester 2016 1. Please write your 1- or 2-digit exam number on

More information

Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures

Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures 2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon

More information

1 Holomorphic functions

1 Holomorphic functions Robert Oeckl CA NOTES 1 15/09/2009 1 1 Holomorphic functions 11 The complex derivative The basic objects of complex analysis are the holomorphic functions These are functions that posses a complex derivative

More information

Solutions to Complex Analysis Prelims Ben Strasser

Solutions to Complex Analysis Prelims Ben Strasser Solutions to Complex Analysis Prelims Ben Strasser In preparation for the complex analysis prelim, I typed up solutions to some old exams. This document includes complete solutions to both exams in 23,

More information

Complex Analysis Qualifying Exam Solutions

Complex Analysis Qualifying Exam Solutions Complex Analysis Qualifying Exam Solutions May, 04 Part.. Let log z be the principal branch of the logarithm defined on G = {z C z (, 0]}. Show that if t > 0, then the equation log z = t has exactly one

More information

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II

NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part II Chapter 2 Further properties of analytic functions 21 Local/Global behavior of analytic functions;

More information

Proposition 5. Group composition in G 1 (N) induces the structure of an abelian group on K 1 (X):

Proposition 5. Group composition in G 1 (N) induces the structure of an abelian group on K 1 (X): 2 RICHARD MELROSE 3. Lecture 3: K-groups and loop groups Wednesday, 3 September, 2008 Reconstructed, since I did not really have notes { because I was concentrating too hard on the 3 lectures on blow-up

More information

The weak topology of locally convex spaces and the weak-* topology of their duals

The weak topology of locally convex spaces and the weak-* topology of their duals The weak topology of locally convex spaces and the weak-* topology of their duals Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Introduction These notes

More information

Normed and Banach Spaces

Normed and Banach Spaces (August 30, 2005) Normed and Banach Spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ We have seen that many interesting spaces of functions have natural structures of Banach spaces:

More information

SPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS

SPRING 2006 PRELIMINARY EXAMINATION SOLUTIONS SPRING 006 PRELIMINARY EXAMINATION SOLUTIONS 1A. Let G be the subgroup of the free abelian group Z 4 consisting of all integer vectors (x, y, z, w) such that x + 3y + 5z + 7w = 0. (a) Determine a linearly

More information

An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010

An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 An Introduction to Complex Analysis and Geometry John P. D Angelo, Pure and Applied Undergraduate Texts Volume 12, American Mathematical Society, 2010 John P. D Angelo, Univ. of Illinois, Urbana IL 61801.

More information

Our goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always

Our goal is to solve a general constant coecient linear second order. this way but that will not always happen). Once we have y 1, it will always October 5 Relevant reading: Section 2.1, 2.2, 2.3 and 2.4 Our goal is to solve a general constant coecient linear second order ODE a d2 y dt + bdy + cy = g (t) 2 dt where a, b, c are constants and a 0.

More information

Topic 4 Notes Jeremy Orloff

Topic 4 Notes Jeremy Orloff Topic 4 Notes Jeremy Orloff 4 auchy s integral formula 4. Introduction auchy s theorem is a big theorem which we will use almost daily from here on out. Right away it will reveal a number of interesting

More information

Math 225B: Differential Geometry, Final

Math 225B: Differential Geometry, Final Math 225B: Differential Geometry, Final Ian Coley March 5, 204 Problem Spring 20,. Show that if X is a smooth vector field on a (smooth) manifold of dimension n and if X p is nonzero for some point of

More information

are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication 7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

More information

THE RESIDUE THEOREM. f(z) dz = 2πi res z=z0 f(z). C

THE RESIDUE THEOREM. f(z) dz = 2πi res z=z0 f(z). C THE RESIDUE THEOREM ontents 1. The Residue Formula 1 2. Applications and corollaries of the residue formula 2 3. ontour integration over more general curves 5 4. Defining the logarithm 7 Now that we have

More information

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem 56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi

More information

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition) Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

More information

An introduction to some aspects of functional analysis

An introduction to some aspects of functional analysis An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms

More information

Counterexamples to witness conjectures. Notations Let E be the set of admissible constant expressions built up from Z; + ;?;;/; exp and log. Here a co

Counterexamples to witness conjectures. Notations Let E be the set of admissible constant expressions built up from Z; + ;?;;/; exp and log. Here a co Counterexamples to witness conjectures Joris van der Hoeven D pt. de Math matiques (b t. 45) Universit Paris-Sud 91405 Orsay CEDEX France August 15, 003 Consider the class of exp-log constants, which is

More information

Course 214 Basic Properties of Holomorphic Functions Second Semester 2008

Course 214 Basic Properties of Holomorphic Functions Second Semester 2008 Course 214 Basic Properties of Holomorphic Functions Second Semester 2008 David R. Wilkins Copyright c David R. Wilkins 1989 2008 Contents 7 Basic Properties of Holomorphic Functions 72 7.1 Taylor s Theorem

More information

Riemann Surfaces and Algebraic Curves

Riemann Surfaces and Algebraic Curves Riemann Surfaces and Algebraic Curves JWR Tuesday December 11, 2001, 9:03 AM We describe the relation between algebraic curves and Riemann surfaces. An elementary reference for this material is [1]. 1

More information

NOTES ON RIEMANN S ZETA FUNCTION. Γ(z) = t z 1 e t dt

NOTES ON RIEMANN S ZETA FUNCTION. Γ(z) = t z 1 e t dt NOTES ON RIEMANN S ZETA FUNCTION DRAGAN MILIČIĆ. Gamma function.. Definition of the Gamma function. The integral Γz = t z e t dt is well-defined and defines a holomorphic function in the right half-plane

More information

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea

(1.) For any subset P S we denote by L(P ) the abelian group of integral relations between elements of P, i.e. L(P ) := ker Z P! span Z P S S : For ea Torsion of dierentials on toric varieties Klaus Altmann Institut fur reine Mathematik, Humboldt-Universitat zu Berlin Ziegelstr. 13a, D-10099 Berlin, Germany. E-mail: altmann@mathematik.hu-berlin.de Abstract

More information

Problem Set 5 Solution Set

Problem Set 5 Solution Set Problem Set 5 Solution Set Anthony Varilly Math 113: Complex Analysis, Fall 2002 1. (a) Let g(z) be a holomorphic function in a neighbourhood of z = a. Suppose that g(a) = 0. Prove that g(z)/(z a) extends

More information

A note on a construction of J. F. Feinstein

A note on a construction of J. F. Feinstein STUDIA MATHEMATICA 169 (1) (2005) A note on a construction of J. F. Feinstein by M. J. Heath (Nottingham) Abstract. In [6] J. F. Feinstein constructed a compact plane set X such that R(X), the uniform

More information

Congurations of periodic orbits for equations with delayed positive feedback

Congurations of periodic orbits for equations with delayed positive feedback Congurations of periodic orbits for equations with delayed positive feedback Dedicated to Professor Tibor Krisztin on the occasion of his 60th birthday Gabriella Vas 1 MTA-SZTE Analysis and Stochastics

More information

1. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials.

1. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials A rational function / is a quotient of two polynomials P, Q with 0 By Fundamental Theorem of algebra, =

More information

Problem 1A. Calculus. Problem 3A. Real analysis. f(x) = 0 x = 0.

Problem 1A. Calculus. Problem 3A. Real analysis. f(x) = 0 x = 0. Problem A. Calculus Find the length of the spiral given in polar coordinates by r = e θ, < θ 0. Solution: The length is 0 θ= ds where by Pythagoras ds = dr 2 + (rdθ) 2 = dθ 2e θ, so the length is 0 2e

More information

4 Uniform convergence

4 Uniform convergence 4 Uniform convergence In the last few sections we have seen several functions which have been defined via series or integrals. We now want to develop tools that will allow us to show that these functions

More information

Richard DiSalvo. Dr. Elmer. Mathematical Foundations of Economics. Fall/Spring,

Richard DiSalvo. Dr. Elmer. Mathematical Foundations of Economics. Fall/Spring, The Finite Dimensional Normed Linear Space Theorem Richard DiSalvo Dr. Elmer Mathematical Foundations of Economics Fall/Spring, 20-202 The claim that follows, which I have called the nite-dimensional normed

More information

RIEMANN S INEQUALITY AND RIEMANN-ROCH

RIEMANN S INEQUALITY AND RIEMANN-ROCH RIEMANN S INEQUALITY AND RIEMANN-ROCH DONU ARAPURA Fix a compact connected Riemann surface X of genus g. Riemann s inequality gives a sufficient condition to construct meromorphic functions with prescribed

More information

Complex Analysis Problems

Complex Analysis Problems Complex Analysis Problems transcribed from the originals by William J. DeMeo October 2, 2008 Contents 99 November 2 2 2 200 November 26 4 3 2006 November 3 6 4 2007 April 6 7 5 2007 November 6 8 99 NOVEMBER

More information

THIRD SEMESTER M. Sc. DEGREE (MATHEMATICS) EXAMINATION (CUSS PG 2010) MODEL QUESTION PAPER MT3C11: COMPLEX ANALYSIS

THIRD SEMESTER M. Sc. DEGREE (MATHEMATICS) EXAMINATION (CUSS PG 2010) MODEL QUESTION PAPER MT3C11: COMPLEX ANALYSIS THIRD SEMESTER M. Sc. DEGREE (MATHEMATICS) EXAMINATION (CUSS PG 2010) MODEL QUESTION PAPER MT3C11: COMPLEX ANALYSIS TIME:3 HOURS Maximum weightage:36 PART A (Short Answer Type Question 1-14) Answer All

More information

THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOZZI AND M. PUGH.

THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOZZI AND M. PUGH. THE LUBRICATION APPROXIMATION FOR THIN VISCOUS FILMS: REGULARITY AND LONG TIME BEHAVIOR OF WEAK SOLUTIONS A.L. BERTOI AND M. PUGH April 1994 Abstract. We consider the fourth order degenerate diusion equation

More information

Remarks on multifunction-based. dynamical systems. Bela Vizvari c. RRR , July, 2001

Remarks on multifunction-based. dynamical systems. Bela Vizvari c. RRR , July, 2001 R u t c o r Research R e p o r t Remarks on multifunction-based dynamical systems Gergely KOV ACS a Bela Vizvari c Marian MURESAN b RRR 43-2001, July, 2001 RUTCOR Rutgers Center for Operations Research

More information

5 Introduction to Complex Dynamics

5 Introduction to Complex Dynamics Dynamics, Chaos, and Fractals (part 5): Introduction to Complex Dynamics (by Evan Dummit, 015, v. 1.0) Contents 5 Introduction to Complex Dynamics 1 5.1 Dynamical Properties of Complex-Valued Functions...........................

More information

COMPLEX ANALYSIS Spring 2014

COMPLEX ANALYSIS Spring 2014 COMPLEX ANALYSIS Spring 24 Homework 4 Solutions Exercise Do and hand in exercise, Chapter 3, p. 4. Solution. The exercise states: Show that if a

More information

Primes in arithmetic progressions

Primes in arithmetic progressions (September 26, 205) Primes in arithmetic progressions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/mfms/notes 205-6/06 Dirichlet.pdf].

More information

ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2.

ANALYSIS QUALIFYING EXAM FALL 2017: SOLUTIONS. 1 cos(nx) lim. n 2 x 2. g n (x) = 1 cos(nx) n 2 x 2. x 2. ANALYSIS QUALIFYING EXAM FALL 27: SOLUTIONS Problem. Determine, with justification, the it cos(nx) n 2 x 2 dx. Solution. For an integer n >, define g n : (, ) R by Also define g : (, ) R by g(x) = g n

More information

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space

PROBLEMS. (b) (Polarization Identity) Show that in any inner product space 1 Professor Carl Cowen Math 54600 Fall 09 PROBLEMS 1. (Geometry in Inner Product Spaces) (a) (Parallelogram Law) Show that in any inner product space x + y 2 + x y 2 = 2( x 2 + y 2 ). (b) (Polarization

More information

Riemann sphere and rational maps

Riemann sphere and rational maps Chapter 3 Riemann sphere and rational maps 3.1 Riemann sphere It is sometimes convenient, and fruitful, to work with holomorphic (or in general continuous) functions on a compact space. However, we wish

More information

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

More information

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, ) II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups

More information

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1. Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

More information

S. Marmi and D. Sauzin CONTENTS. Introduction 2. Monogenic properties of the solutions of the cohomological equation 2. C -holomorphic and C -holomorp

S. Marmi and D. Sauzin CONTENTS. Introduction 2. Monogenic properties of the solutions of the cohomological equation 2. C -holomorphic and C -holomorp Quasianalytic monogenic solutions of a cohomological equation 8 December 2000 Stefano Marmi and David Sauzin 2 Abstract We prove that the solutions of a cohomological equation of complex dimension one

More information

Exercise Solutions to Functional Analysis

Exercise Solutions to Functional Analysis Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n

More information

g(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0)

g(x) = P (y) Proof. This is true for n = 0. Assume by the inductive hypothesis that g (n) (0) = 0 for some n. Compute g (n) (h) g (n) (0) Mollifiers and Smooth Functions We say a function f from C is C (or simply smooth) if all its derivatives to every order exist at every point of. For f : C, we say f is C if all partial derivatives to

More information

CHAPTER 1 Basic properties of holomorphic functions Preview of differences between one and several variables

CHAPTER 1 Basic properties of holomorphic functions Preview of differences between one and several variables CHAPTER 1 Basic properties of holomorphic functions Preview of differences between one and several variables For any n 1, the holomorphy or complex differentiability of a function on a domain in C n implies

More information

Complex Analysis Homework 4: Solutions

Complex Analysis Homework 4: Solutions Complex Analysis Fall 2007 Homework 4: Solutions 1.5.2. a The function fz 3z 2 +7z+5 is a polynomial so is analytic everywhere with derivative f z 6z + 7. b The function fz 2z + 3 4 is a composition of

More information

INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

More information

Introduction to Real Analysis

Introduction to Real Analysis Introduction to Real Analysis Joshua Wilde, revised by Isabel Tecu, Takeshi Suzuki and María José Boccardi August 13, 2013 1 Sets Sets are the basic objects of mathematics. In fact, they are so basic that

More information

From the definition of a surface, each point has a neighbourhood U and a homeomorphism. U : ϕ U(U U ) ϕ U (U U )

From the definition of a surface, each point has a neighbourhood U and a homeomorphism. U : ϕ U(U U ) ϕ U (U U ) 3 Riemann surfaces 3.1 Definitions and examples From the definition of a surface, each point has a neighbourhood U and a homeomorphism ϕ U from U to an open set V in R 2. If two such neighbourhoods U,

More information

On probability measures with algebraic Cauchy transform

On probability measures with algebraic Cauchy transform On probability measures with algebraic Cauchy transform Boris Shapiro Stockholm University IMS Singapore, January 8, 2014 Basic definitions The logarithimic potential and the Cauchy transform of a (compactly

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

SYMPLECTIC LEFSCHETZ FIBRATIONS ALEXANDER CAVIEDES CASTRO

SYMPLECTIC LEFSCHETZ FIBRATIONS ALEXANDER CAVIEDES CASTRO SYMPLECTIC LEFSCHETZ FIBRATIONS ALEXANDER CAVIEDES CASTRO. Introduction A Lefschetz pencil is a construction that comes from algebraic geometry, but it is closely related with symplectic geometry. Indeed,

More information

s P = f(ξ n )(x i x i 1 ). i=1

s P = f(ξ n )(x i x i 1 ). i=1 Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological

More information

Complex Analysis Math 220C Spring 2008

Complex Analysis Math 220C Spring 2008 Complex Analysis Math 220C Spring 2008 Bernard Russo June 2, 2008 Contents 1 Monday March 31, 2008 class cancelled due to the Master s travel plans 1 2 Wednesday April 2, 2008 Course information; Riemann

More information

ADDENDUM B: CONSTRUCTION OF R AND THE COMPLETION OF A METRIC SPACE

ADDENDUM B: CONSTRUCTION OF R AND THE COMPLETION OF A METRIC SPACE ADDENDUM B: CONSTRUCTION OF R AND THE COMPLETION OF A METRIC SPACE ANDREAS LEOPOLD KNUTSEN Abstract. These notes are written as supplementary notes for the course MAT11- Real Analysis, taught at the University

More information

balls, Edalat and Heckmann [4] provided a simple explicit construction of a computational model for a Polish space. They also gave domain theoretic pr

balls, Edalat and Heckmann [4] provided a simple explicit construction of a computational model for a Polish space. They also gave domain theoretic pr Electronic Notes in Theoretical Computer Science 6 (1997) URL: http://www.elsevier.nl/locate/entcs/volume6.html 9 pages Computational Models for Ultrametric Spaces Bob Flagg Department of Mathematics and

More information

Complex numbers, the exponential function, and factorization over C

Complex numbers, the exponential function, and factorization over C Complex numbers, the exponential function, and factorization over C 1 Complex Numbers Recall that for every non-zero real number x, its square x 2 = x x is always positive. Consequently, R does not contain

More information

Supplementary Notes for W. Rudin: Principles of Mathematical Analysis

Supplementary Notes for W. Rudin: Principles of Mathematical Analysis Supplementary Notes for W. Rudin: Principles of Mathematical Analysis SIGURDUR HELGASON In 8.00B it is customary to cover Chapters 7 in Rudin s book. Experience shows that this requires careful planning

More information

Micro-support of sheaves

Micro-support of sheaves Micro-support of sheaves Vincent Humilière 17/01/14 The microlocal theory of sheaves and in particular the denition of the micro-support is due to Kashiwara and Schapira (the main reference is their book

More information

290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f

290 J.M. Carnicer, J.M. Pe~na basis (u 1 ; : : : ; u n ) consisting of minimally supported elements, yet also has a basis (v 1 ; : : : ; v n ) which f Numer. Math. 67: 289{301 (1994) Numerische Mathematik c Springer-Verlag 1994 Electronic Edition Least supported bases and local linear independence J.M. Carnicer, J.M. Pe~na? Departamento de Matematica

More information

Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations.

Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differenti. equations. Lecture 9 Metric spaces. The contraction fixed point theorem. The implicit function theorem. The existence of solutions to differential equations. 1 Metric spaces 2 Completeness and completion. 3 The contraction

More information

Continuity of convex functions in normed spaces

Continuity of convex functions in normed spaces Continuity of convex functions in normed spaces In this chapter, we consider continuity properties of real-valued convex functions defined on open convex sets in normed spaces. Recall that every infinitedimensional

More information

Formal Groups. Niki Myrto Mavraki

Formal Groups. Niki Myrto Mavraki Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE

FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE FUNCTIONAL ANALYSIS LECTURE NOTES: WEAK AND WEAK* CONVERGENCE CHRISTOPHER HEIL 1. Weak and Weak* Convergence of Vectors Definition 1.1. Let X be a normed linear space, and let x n, x X. a. We say that

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality

121B: ALGEBRAIC TOPOLOGY. Contents. 6. Poincaré Duality 121B: ALGEBRAIC TOPOLOGY Contents 6. Poincaré Duality 1 6.1. Manifolds 2 6.2. Orientation 3 6.3. Orientation sheaf 9 6.4. Cap product 11 6.5. Proof for good coverings 15 6.6. Direct limit 18 6.7. Proof

More information

Complex Analysis review notes for weeks 1-6

Complex Analysis review notes for weeks 1-6 Complex Analysis review notes for weeks -6 Peter Milley Semester 2, 2007 In what follows, unless stated otherwise a domain is a connected open set. Generally we do not include the boundary of the set,

More information

CONSEQUENCES OF POWER SERIES REPRESENTATION

CONSEQUENCES OF POWER SERIES REPRESENTATION CONSEQUENCES OF POWER SERIES REPRESENTATION 1. The Uniqueness Theorem Theorem 1.1 (Uniqueness). Let Ω C be a region, and consider two analytic functions f, g : Ω C. Suppose that S is a subset of Ω that

More information

The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria On the Foundations of Nonlinear Generalized Functions I E. Farkas M. Grosser M. Kunzinger

More information

Independence of Boolean algebras and forcing

Independence of Boolean algebras and forcing Annals of Pure and Applied Logic 124 (2003) 179 191 www.elsevier.com/locate/apal Independence of Boolean algebras and forcing Milos S. Kurilic Department of Mathematics and Informatics, University of Novi

More information

MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

More information

University of California. Berkeley, CA fzhangjun johans lygeros Abstract

University of California. Berkeley, CA fzhangjun johans lygeros Abstract Dynamical Systems Revisited: Hybrid Systems with Zeno Executions Jun Zhang, Karl Henrik Johansson y, John Lygeros, and Shankar Sastry Department of Electrical Engineering and Computer Sciences University

More information

2. Complex Analytic Functions

2. Complex Analytic Functions 2. Complex Analytic Functions John Douglas Moore July 6, 2011 Recall that if A and B are sets, a function f : A B is a rule which assigns to each element a A a unique element f(a) B. In this course, we

More information

Continuity. Chapter 4

Continuity. Chapter 4 Chapter 4 Continuity Throughout this chapter D is a nonempty subset of the real numbers. We recall the definition of a function. Definition 4.1. A function from D into R, denoted f : D R, is a subset of

More information

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain.

After taking the square and expanding, we get x + y 2 = (x + y) (x + y) = x 2 + 2x y + y 2, inequality in analysis, we obtain. Lecture 1: August 25 Introduction. Topology grew out of certain questions in geometry and analysis about 100 years ago. As Wikipedia puts it, the motivating insight behind topology is that some geometric

More information

What to remember about metric spaces

What to remember about metric spaces Division of the Humanities and Social Sciences What to remember about metric spaces KC Border These notes are (I hope) a gentle introduction to the topological concepts used in economic theory. If the

More information

RIEMANN SURFACES. ω = ( f i (γ(t))γ i (t))dt.

RIEMANN SURFACES. ω = ( f i (γ(t))γ i (t))dt. RIEMANN SURFACES 6. Week 7: Differential forms. De Rham complex 6.1. Introduction. The notion of differential form is important for us for various reasons. First of all, one can integrate a k-form along

More information

Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert. f(x) = f + (x) + f (x).

Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert. f(x) = f + (x) + f (x). References: Notes on Integrable Functions and the Riesz Representation Theorem Math 8445, Winter 06, Professor J. Segert Evans, Partial Differential Equations, Appendix 3 Reed and Simon, Functional Analysis,

More information

ENTRY POTENTIAL THEORY. [ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: T. Ransford, Potential theory in the complex plane.

ENTRY POTENTIAL THEORY. [ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: T. Ransford, Potential theory in the complex plane. ENTRY POTENTIAL THEORY [ENTRY POTENTIAL THEORY] Authors: Oliver Knill: jan 2003 Literature: T. Ransford, Potential theory in the complex plane. Analytic [Analytic] Let D C be an open set. A continuous

More information

Integration and Manifolds

Integration and Manifolds Integration and Manifolds Course No. 100 311 Fall 2007 Michael Stoll Contents 1. Manifolds 2 2. Differentiable Maps and Tangent Spaces 8 3. Vector Bundles and the Tangent Bundle 13 4. Orientation and Orientability

More information

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients

More information

Complex Analysis Homework 9: Solutions

Complex Analysis Homework 9: Solutions Complex Analysis Fall 2007 Homework 9: Solutions 3..4 (a) Let z C \ {ni : n Z}. Then /(n 2 + z 2 ) n /n 2 n 2 n n 2 + z 2. According to the it comparison test from calculus, the series n 2 + z 2 converges

More information

Proof of a simple case of the Siegel-Weil formula. 1. Weil/oscillator representations

Proof of a simple case of the Siegel-Weil formula. 1. Weil/oscillator representations (March 6, 2014) Proof of a simple case of the Siegel-Weil formula Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ First, I confess I never understood Siegel s arguments for his mass

More information

TOPOLOGY TAKE-HOME CLAY SHONKWILER

TOPOLOGY TAKE-HOME CLAY SHONKWILER TOPOLOGY TAKE-HOME CLAY SHONKWILER 1. The Discrete Topology Let Y = {0, 1} have the discrete topology. Show that for any topological space X the following are equivalent. (a) X has the discrete topology.

More information

Hilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality

Hilbert spaces. 1. Cauchy-Schwarz-Bunyakowsky inequality (October 29, 2016) Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/notes 2016-17/03 hsp.pdf] Hilbert spaces are

More information

Introduction to Topology

Introduction to Topology Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about

More information

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2.

(x 1, y 1 ) = (x 2, y 2 ) if and only if x 1 = x 2 and y 1 = y 2. 1. Complex numbers A complex number z is defined as an ordered pair z = (x, y), where x and y are a pair of real numbers. In usual notation, we write z = x + iy, where i is a symbol. The operations of

More information

Fuchsian groups. 2.1 Definitions and discreteness

Fuchsian groups. 2.1 Definitions and discreteness 2 Fuchsian groups In the previous chapter we introduced and studied the elements of Mob(H), which are the real Moebius transformations. In this chapter we focus the attention of special subgroups of this

More information

f(x) f(z) c x z > 0 1

f(x) f(z) c x z > 0 1 INVERSE AND IMPLICIT FUNCTION THEOREMS I use df x for the linear transformation that is the differential of f at x.. INVERSE FUNCTION THEOREM Definition. Suppose S R n is open, a S, and f : S R n is a

More information

NATIONAL UNIVERSITY OF SINGAPORE. Department of Mathematics. MA4247 Complex Analysis II. Lecture Notes Part IV

NATIONAL UNIVERSITY OF SINGAPORE. Department of Mathematics. MA4247 Complex Analysis II. Lecture Notes Part IV NATIONAL UNIVERSITY OF SINGAPORE Department of Mathematics MA4247 Complex Analysis II Lecture Notes Part IV Chapter 2. Further properties of analytic/holomorphic functions (continued) 2.4. The Open mapping

More information

M4P52 Manifolds, 2016 Problem Sheet 1

M4P52 Manifolds, 2016 Problem Sheet 1 Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete

More information

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions

Economics 204 Fall 2011 Problem Set 2 Suggested Solutions Economics 24 Fall 211 Problem Set 2 Suggested Solutions 1. Determine whether the following sets are open, closed, both or neither under the topology induced by the usual metric. (Hint: think about limit

More information

Extreme points of compact convex sets

Extreme points of compact convex sets Extreme points of compact convex sets In this chapter, we are going to show that compact convex sets are determined by a proper subset, the set of its extreme points. Let us start with the main definition.

More information

REAL AND COMPLEX ANALYSIS

REAL AND COMPLEX ANALYSIS REAL AND COMPLE ANALYSIS Third Edition Walter Rudin Professor of Mathematics University of Wisconsin, Madison Version 1.1 No rights reserved. Any part of this work can be reproduced or transmitted in any

More information