Glossary

3.2 Euler-Maclaurin summation formula

3 min read•august 9, 2024

The bridges the gap between discrete sums and continuous integrals. It's a powerful tool that expresses sums as integrals plus correction terms, allowing for more accurate approximations and insights into series behavior.

This formula is crucial in analytic number theory, connecting discrete mathematics to continuous analysis. It's used to study important mathematical objects like the and , and has applications in physics and computer science.

Euler-Maclaurin Formula and Components

Formula Structure and Integral Approximation

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  • Euler-Maclaurin formula bridges discrete sums and continuous integrals
  • Expresses sum of a function over an interval as integral plus correction terms
  • General form: ∑k=abf(k)=∫abf(x)dx+f(a)+f(b)2+∑k=1mBk(2k)![f(2k−1)(b)−f(2k−1)(a)]+Rm\sum_{k=a}^b f(k) = \int_a^b f(x) dx + \frac{f(a) + f(b)}{2} + \sum_{k=1}^m \frac{B_k}{(2k)!} [f^{(2k-1)}(b) - f^{(2k-1)}(a)] + R_m
  • Integral approximation represents continuous analog of discrete sum
  • Correction terms adjust for discretization errors
  • Formula applies to functions with continuous derivatives

Remainder Term and Asymptotic Expansion

  • Remainder term RmR_m accounts for approximation error
  • Expressed as integral involving
  • Remainder form: Rm=(−1)m+1∫abBm({x})m!f(2m)(x)dxR_m = (-1)^{m+1} \int_a^b \frac{B_m(\{x\})}{m!} f^{(2m)}(x) dx
  • {x}\{x\} denotes fractional part of x
  • Asymptotic expansion obtained by taking m→∞m \to \infty
  • Expansion provides increasingly accurate approximations
  • Convergence depends on function's smoothness and decay properties

Practical Applications and Computational Aspects

  • Used for numerical integration and series acceleration
  • Improves accuracy of numerical summation methods
  • Applies in various fields (physics, engineering, statistics)
  • Computational implementation requires careful handling of truncation errors
  • Effective for slowly converging series (harmonic series)
  • Provides insight into analytic properties of discrete sums

Bernoulli Numbers and Polynomials

Bernoulli Numbers: Definition and Properties

  • Sequence of rational numbers crucial in number theory and analysis
  • Defined recursively or through generating function
  • Generating function: tet−1=∑n=0∞Bntnn!\frac{t}{e^t - 1} = \sum_{n=0}^{\infty} B_n \frac{t^n}{n!}
  • First few values: B0=1B_0 = 1, B1=−12B_1 = -\frac{1}{2}, B2=16B_2 = \frac{1}{6}, B4=−130B_4 = -\frac{1}{30}
  • Odd-indexed (except B1B_1) equal zero
  • Appear in expansions of trigonometric and hyperbolic functions
  • Satisfy important identities (sum of powers formula)
  • Connect to zeta function values at even integers

Bernoulli Polynomials: Characteristics and Relationships

  • Generalization of Bernoulli numbers to polynomial form
  • Defined by generating function: textet−1=∑n=0∞Bn(x)tnn!\frac{te^{xt}}{e^t - 1} = \sum_{n=0}^{\infty} B_n(x) \frac{t^n}{n!}
  • Bn(0)B_n(0) equals the nth Bernoulli number
  • Satisfy differential equation: Bn′(x)=nBn−1(x)B'_n(x) = nB_{n-1}(x)
  • Periodic with period 1: Bn(x+1)−Bn(x)=nxn−1B_n(x+1) - B_n(x) = nx^{n-1}
  • Used in Euler-Maclaurin formula's remainder term
  • Appear in various mathematical identities and formulas
  • Connect Bernoulli numbers to values of polynomials at specific points

Harmonic Series and Euler's Constant

  • Harmonic series defined as ∑n=1∞1n\sum_{n=1}^{\infty} \frac{1}{n}
  • Diverges but grows very slowly
  • Euler-Maclaurin formula provides asymptotic expansion
  • Expansion: Hn∼ln⁡(n)+γ+12n−112n2+1120n4−...H_n \sim \ln(n) + \gamma + \frac{1}{2n} - \frac{1}{12n^2} + \frac{1}{120n^4} - ...
  • γ\gamma represents (approximately 0.5772156649)
  • Euler's constant connects harmonic series to natural logarithm
  • Applications in physics (entropy), computer science (analysis of algorithms)

Riemann Zeta Function and Analytic Continuation

  • Riemann zeta function defined as ζ(s)=∑n=1∞1ns\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
  • Euler-Maclaurin formula enables to complex plane
  • Provides representation valid for all s≠1s \neq 1
  • Reveals functional equation relating ζ(s)\zeta(s) to ζ(1−s)\zeta(1-s)
  • Critical strip 0<ℜ(s)<10 < \Re(s) < 1 contains non-trivial zeros
  • Riemann Hypothesis conjectures all non-trivial zeros have real part 12\frac{1}{2}
  • Connections to prime number distribution and analytic number theory

Stirling's Approximation and Factorial Asymptotics

  • estimates factorial for large n
  • Derived using Euler-Maclaurin formula
  • Main term: n!∼2πn(ne)nn! \sim \sqrt{2\pi n} (\frac{n}{e})^n
  • Full asymptotic expansion: n!∼2πn(ne)n(1+112n+1288n2−13951840n3+...)n! \sim \sqrt{2\pi n} (\frac{n}{e})^n (1 + \frac{1}{12n} + \frac{1}{288n^2} - \frac{139}{51840n^3} + ...)
  • Provides accurate estimates even for moderate values of n
  • Used in probability theory, statistical mechanics, and combinatorics
  • Generalizes to for non-integer arguments

Key Terms to Review (22)

Absolute convergence: Absolute convergence refers to a property of a series where the series of the absolute values of its terms converges. When a series converges absolutely, it implies that the original series converges as well, making absolute convergence a stronger condition than ordinary convergence. This concept is crucial in analyzing the behavior of Dirichlet series, such as L-functions and the Riemann zeta function, as it ensures stability under rearrangement of terms and provides insights into their analytic properties.
Analytic Continuation: Analytic continuation is a technique in complex analysis that extends the domain of a given analytic function beyond its original area of definition, allowing it to be expressed in a broader context. This process is crucial for understanding functions like the Riemann zeta function and Dirichlet L-functions, as it reveals their behavior and properties in different regions of the complex plane.
Asymptotic expansions: Asymptotic expansions are mathematical expressions that provide an approximation of a function in terms of simpler functions as an argument approaches a particular limit, often infinity. They help to analyze the behavior of functions and can be particularly useful in estimating sums or integrals that are difficult to compute directly, thereby connecting to concepts like series approximations and error estimation.
Bernoulli numbers: Bernoulli numbers are a sequence of rational numbers that play a crucial role in number theory, particularly in the evaluation of sums of powers of integers and in the expansion of certain functions. These numbers are deeply connected to the Riemann zeta function, appear in the Euler-Maclaurin summation formula, and help in deriving various identities and formulas related to series and calculus.
Bernoulli Polynomials: Bernoulli polynomials are a sequence of polynomials that are crucial in number theory and mathematical analysis, defined using the generating function $$ rac{t}{e^t - 1} = rac{1}{2} + rac{B_1}{1!}t + rac{B_2}{2!}t^2 + rac{B_3}{3!}t^3 + ...$$ where the coefficients are the Bernoulli numbers. They are particularly important in the Euler-Maclaurin summation formula, which relates sums and integrals, making them useful for approximating sums of functions.
Boundary corrections: Boundary corrections refer to adjustments made in numerical analysis to improve the accuracy of approximations, particularly when summing a function over a discrete range. These corrections help account for the errors that arise when estimating integrals or sums at the endpoints, enhancing the precision of results derived from the Euler-Maclaurin summation formula.
Carl Friedrich Gauss: Carl Friedrich Gauss was a German mathematician and astronomer who made significant contributions to various fields, including number theory, statistics, and astronomy. Known as the 'Prince of Mathematicians,' his work laid foundational principles that are crucial for understanding concepts related to arithmetic functions, prime distribution, and analytic techniques.
Conditional Convergence: Conditional convergence refers to the behavior of an infinite series where it converges, but if the order of its terms is rearranged, it may converge to a different value or even diverge. This property is particularly significant in the context of certain series related to analytic functions, as it highlights the delicate balance between convergence and the ordering of terms.
Dirichlet series: A Dirichlet series is a type of infinite series of the form $$D(s) = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ where $a_n$ are complex coefficients and $s$ is a complex variable. These series are a powerful tool in analytic number theory, linking properties of numbers with functions, especially through their relationship with zeta functions and multiplicative functions.
Error Term: The error term refers to the difference between an estimated quantity and its actual value, often arising in analytic number theory when approximating functions like the prime counting function. In various proofs and formulas, the error term helps quantify how close an approximation is to the true result, providing insight into the accuracy of estimates and the behavior of mathematical functions.
Estimates for sums: Estimates for sums are mathematical techniques used to approximate the value of a sum, often when dealing with infinite series or large finite sums. These estimates help in analyzing the behavior of sums in number theory and are particularly useful in understanding convergence, divergence, and the relationship between summation and integration, especially when using advanced tools like the Euler-Maclaurin summation formula.
Euler-Maclaurin Summation Formula: The Euler-Maclaurin summation formula is a powerful tool in mathematics that connects sums to integrals, providing a way to approximate the sum of a function at discrete points through its integral representation. This formula is particularly useful for estimating the sum of sequences and has applications in number theory, particularly when analyzing series and their convergence properties.
Euler's Constant: Euler's Constant, denoted as $eta$, is a mathematical constant that arises in the analysis of the harmonic series and is defined as the limiting difference between the harmonic series and the natural logarithm. It connects deeply to various mathematical fields, particularly in approximating sums and analyzing the behavior of arithmetic functions, which can be pivotal in understanding their average orders.
Fourier Series: A Fourier series is a way to represent a function as a sum of sinusoidal components, specifically sines and cosines. It breaks down periodic functions into their constituent frequencies, making it useful for analyzing waveforms and solving differential equations. By expressing a function in this manner, it allows for easier manipulation and understanding of its properties, especially in terms of convergence and approximation.
Gamma Function: The Gamma function is a mathematical function that extends the concept of factorials to complex and real number arguments. It is defined for positive real numbers and is denoted as $$ ext{Γ}(n) = (n-1)!$$ for natural numbers. The Gamma function plays a crucial role in various areas of mathematics, including number theory, calculus, and complex analysis, often connecting to concepts like the Riemann zeta function and analytic continuation.
Harmonic Series: The harmonic series is the infinite series formed by the sum of the reciprocals of the positive integers, represented mathematically as $$H_n = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + ... + \frac{1}{n}$$. This series diverges, meaning that as n approaches infinity, the sum grows without bound. It is essential in understanding asymptotic behavior in various mathematical contexts and plays a crucial role in approximating sums via techniques such as the Euler-Maclaurin summation formula and in analyzing the average order of arithmetic functions.
Integration by parts: Integration by parts is a technique used to integrate the product of two functions. It is based on the product rule of differentiation and provides a method for transforming the integral of a product into a more manageable form. This technique is especially useful in contexts like summation formulas, where it can simplify complex sums into integrals, making them easier to evaluate or approximate.
Leonhard Euler: Leonhard Euler was an influential Swiss mathematician and physicist known for his pioneering work in various areas of mathematics, including number theory, graph theory, and calculus. His contributions laid the groundwork for many modern mathematical concepts, including the Riemann zeta function, which connects deeply with analytic number theory and has significant implications in both pure and applied mathematics.
Riemann sums: Riemann sums are a method for approximating the integral of a function by dividing the region under the curve into rectangles, summing their areas to estimate the total area. This approach provides a way to understand the concept of integration by breaking down complex functions into simpler, manageable parts. The idea is foundational in calculus and plays a critical role in more advanced mathematical concepts, particularly in relation to numerical analysis and the Euler-Maclaurin summation formula.
Riemann zeta function: The Riemann zeta function is a complex function defined for complex numbers, which plays a pivotal role in number theory, particularly in understanding the distribution of prime numbers. It is intimately connected to various aspects of analytic number theory, including the functional equation, Dirichlet series, and the famous Riemann Hypothesis that conjectures all non-trivial zeros of the function lie on the critical line in the complex plane.
Stirling's Approximation: Stirling's approximation is a formula used to estimate the factorial of a large number, providing an asymptotic approximation that simplifies calculations involving factorials. It expresses the factorial in terms of simpler functions, making it particularly useful in various areas of mathematics, including the analysis of series and the study of analytic properties of functions like the zeta function. The approximation reveals connections to logarithmic and exponential functions, especially when analyzing growth rates and convergence behaviors.
Taylor series: A Taylor series is a mathematical representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It provides a powerful tool for approximating functions using polynomials, which can simplify complex calculations and help analyze the behavior of functions near a given point. The Taylor series plays a significant role in various fields, including calculus, numerical analysis, and is particularly useful in understanding the Euler-Maclaurin summation formula.
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