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Papers by Joseph Bak

Research paper thumbnail of From Repeated Tosses of a Fair Die to the Renewal Theorem

Albanian journal of mathematics, Dec 14, 2021

Research paper thumbnail of Complex Analysis

Research paper thumbnail of The rate of approximation by means of polynomials with restricted coefficients

Israel Journal of Mathematics, 1977

ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ... more ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ,n = 1,2, …, ¦a k ≦ Akk}. We consider the rate of approximation by elements ofP A , of continuous functions in [0, 1] which vanish at x = 0. Also a classP A is called “efficient” if globally it guarantees the Jackson rate of approximation. Some necessary conditions for efficiency and some sufficient ones are derived.

Research paper thumbnail of Complex Analysis -3/E

[Research paper thumbnail of Rational combinations of x k; k 0 are always dense in C[0; 1](https://a.academia-assets.com/images/blank-paper.jpg)

Research paper thumbnail of The linear Diophantine problem of Frobenius

Jp Journal of Algebra Number Theory and Applications, 2005

Research paper thumbnail of The Evolution of Cauchy’s Closed Curve Theorem and Newman’s Simple Proof

The American Mathematical Monthly

Research paper thumbnail of M?ntz-Jackson theorems in , < 2

Research paper thumbnail of Complex Analysis / J. Bak, D.J. Newman

Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Inte... more Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Integrales de línea y funciones enteras. Propiedades de las F. enteras. De las F. Analíticas. Dominios conectado simplemente. Singularidades aisladas de una f. analítica. El teorema de residuo. Aplicación del teorema a la evaluación de sumas de integrales. Contorno de las técnicas integrales. Intruducción al mapeo conformal. El teorema de mapeo de Riemann. Teoremas de módulos máximos de dominios no limitados. Funciones armónicas. Distintas formas de f. analíticas. Continuación analítica, f. gamma y zeta. Aplicaciones.

Research paper thumbnail of Complex analysis. 2nd ed

Research paper thumbnail of Complex analysis / Joseph Bak, Donald J. Newman

ABSTRACT Incluye bibliografía e índice

Research paper thumbnail of The Recreational Gambler: Paying the Price for More Time at the Table

Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune ... more Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune either reaches a predetermined goal or is entirely lost. The probability of success is the probability that he will reach his goal. The duration of play is the expected number of bets until the contest is over. This article deals with the effect on both the probability of success and the duration of play if the size of the individual bets is reduced by an integral factor. In particular, we determine when the duration of play is increased most dramatically, and we obtain an upper bound for the ratio of increased duration.

Research paper thumbnail of Properties of Entire Functions

Undergraduate Texts in Mathematics, 2010

ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} &a... more ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} &amp; {z \ne a} \\{f&#39;(a)} &amp; {z = a} \\ } \right. $ g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} &amp; {z \ne a} \\{f&#39;(a)} &amp; {z = a} \\ \end{array} } \right. then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f. (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.)We begin by showing that the Rectangle Theorem applies to g.

Research paper thumbnail of The Anxious Gambler's Ruin

Mathematics Magazine, 2001

Research paper thumbnail of On the efficiency of general rational approximation

Journal of Approximation Theory, 1977

Research paper thumbnail of Approximation by Müntz-polynomials away from the origin

Journal of Approximation Theory, 1982

Research paper thumbnail of Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Undergraduate Texts in Mathematics, 2010

ABSTRACT In the next section, we will see how various types of (real) definite integrals can be a... more ABSTRACT In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.

Research paper thumbnail of Analytic Continuation; The Gamma and Zeta Functions

Undergraduate Texts in Mathematics, 2010

ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f ca... more ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f can be continued analytically to a region D 1 that intersects D if there exists a function g, analytic in D 1 and such that g = f throughout D1 ÇD2 D_1 \cap D_2 . By the Uniqueness Theorem (6.9) any such continuation of f is uniquely determined. (It is possible, however, to have two analytic continuations g 1 and g 2 of a function f to regions D 1 and D 2 respectively with g1 ¹ g2 g_1 \ne g_2 throughout D1 ÇD2 D_1 \cap D_2 . See Exercise 1.)

Research paper thumbnail of Rational combinations of , ? ? 0 are always dense in 60, 19

Research paper thumbnail of Extremal Points, Critical Points, and Saddle Points of Analytic Functions

The American Mathematical Monthly

This note examines the relation among critical points, saddle points, and extremal points of a co... more This note examines the relation among critical points, saddle points, and extremal points of a complex analytic function and thereby presents an aspect of function theory that is in marked contrast to its well-known real-analytic counterpart. Also shown is an interesting application of this relation to the estimation of certain real sums.

Research paper thumbnail of From Repeated Tosses of a Fair Die to the Renewal Theorem

Albanian journal of mathematics, Dec 14, 2021

Research paper thumbnail of Complex Analysis

Research paper thumbnail of The rate of approximation by means of polynomials with restricted coefficients

Israel Journal of Mathematics, 1977

ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ... more ABSTRACT For a sequenceA = {Ak}k−1∞ of positive constants letP A = {p(x): p(x) = Σ k−1 n a k x k ,n = 1,2, …, ¦a k ≦ Akk}. We consider the rate of approximation by elements ofP A , of continuous functions in [0, 1] which vanish at x = 0. Also a classP A is called “efficient” if globally it guarantees the Jackson rate of approximation. Some necessary conditions for efficiency and some sufficient ones are derived.

Research paper thumbnail of Complex Analysis -3/E

[Research paper thumbnail of Rational combinations of x k; k 0 are always dense in C[0; 1](https://a.academia-assets.com/images/blank-paper.jpg)

Research paper thumbnail of The linear Diophantine problem of Frobenius

Jp Journal of Algebra Number Theory and Applications, 2005

Research paper thumbnail of The Evolution of Cauchy’s Closed Curve Theorem and Newman’s Simple Proof

The American Mathematical Monthly

Research paper thumbnail of M?ntz-Jackson theorems in , < 2

Research paper thumbnail of Complex Analysis / J. Bak, D.J. Newman

Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Inte... more Contenido: Los números complejos. Funciones de la variable compleja Z. Funciones analíticas. Integrales de línea y funciones enteras. Propiedades de las F. enteras. De las F. Analíticas. Dominios conectado simplemente. Singularidades aisladas de una f. analítica. El teorema de residuo. Aplicación del teorema a la evaluación de sumas de integrales. Contorno de las técnicas integrales. Intruducción al mapeo conformal. El teorema de mapeo de Riemann. Teoremas de módulos máximos de dominios no limitados. Funciones armónicas. Distintas formas de f. analíticas. Continuación analítica, f. gamma y zeta. Aplicaciones.

Research paper thumbnail of Complex analysis. 2nd ed

Research paper thumbnail of Complex analysis / Joseph Bak, Donald J. Newman

ABSTRACT Incluye bibliografía e índice

Research paper thumbnail of The Recreational Gambler: Paying the Price for More Time at the Table

Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune ... more Consider a gambler who makes a series of even-money bets of fixed size until his initial fortune either reaches a predetermined goal or is entirely lost. The probability of success is the probability that he will reach his goal. The duration of play is the expected number of bets until the contest is over. This article deals with the effect on both the probability of success and the duration of play if the size of the individual bets is reduced by an integral factor. In particular, we determine when the duration of play is increased most dramatically, and we obtain an upper bound for the ratio of increased duration.

Research paper thumbnail of Properties of Entire Functions

Undergraduate Texts in Mathematics, 2010

ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} &a... more ABSTRACT We now show that if f is entire and if $ g(z) = \left\{ {{*{20}{c}} {f(z) - f(a)} &amp; {z \ne a} \\{f&#39;(a)} &amp; {z = a} \\ } \right. $ g(z) = \left\{ {\begin{array}{*{20}{c}} {f(z) - f(a)} &amp; {z \ne a} \\{f&#39;(a)} &amp; {z = a} \\ \end{array} } \right. then the Integral Theorem (4.15) and Closed Curve Theorem (4.16) apply to g as well as to f. (Note that since f is entire, g is continuous; however, it is not obvious that g is entire.)We begin by showing that the Rectangle Theorem applies to g.

Research paper thumbnail of The Anxious Gambler's Ruin

Mathematics Magazine, 2001

Research paper thumbnail of On the efficiency of general rational approximation

Journal of Approximation Theory, 1977

Research paper thumbnail of Approximation by Müntz-polynomials away from the origin

Journal of Approximation Theory, 1982

Research paper thumbnail of Applications of the Residue Theorem to the Evaluation of Integrals and Sums

Undergraduate Texts in Mathematics, 2010

ABSTRACT In the next section, we will see how various types of (real) definite integrals can be a... more ABSTRACT In the next section, we will see how various types of (real) definite integrals can be associated with integrals around closed curves in the complex plane, so that the Residue Theorem will become a handy tool for definite integration.

Research paper thumbnail of Analytic Continuation; The Gamma and Zeta Functions

Undergraduate Texts in Mathematics, 2010

ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f ca... more ABSTRACT Suppose we are given a function f which is analytic in a region D. We will say that f can be continued analytically to a region D 1 that intersects D if there exists a function g, analytic in D 1 and such that g = f throughout D1 ÇD2 D_1 \cap D_2 . By the Uniqueness Theorem (6.9) any such continuation of f is uniquely determined. (It is possible, however, to have two analytic continuations g 1 and g 2 of a function f to regions D 1 and D 2 respectively with g1 ¹ g2 g_1 \ne g_2 throughout D1 ÇD2 D_1 \cap D_2 . See Exercise 1.)

Research paper thumbnail of Rational combinations of , ? ? 0 are always dense in 60, 19

Research paper thumbnail of Extremal Points, Critical Points, and Saddle Points of Analytic Functions

The American Mathematical Monthly

This note examines the relation among critical points, saddle points, and extremal points of a co... more This note examines the relation among critical points, saddle points, and extremal points of a complex analytic function and thereby presents an aspect of function theory that is in marked contrast to its well-known real-analytic counterpart. Also shown is an interesting application of this relation to the estimation of certain real sums.