Euler-Maclaurin Formula As An Asymptotic Series

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The Euler-Maclaurin formula is a powerful mathematical tool used to approximate the value of a definite integral. It is a summation formula that relates the sum of a function at discrete points to the integral of the function over a continuous interval. In this article, we will explore the Euler-Maclaurin formula as an asymptotic series, discussing its derivation, properties, and applications.

Introduction to the Euler-Maclaurin Formula

The Euler-Maclaurin formula is a generalization of the trapezoidal rule, which is a method for approximating the value of a definite integral. The formula is named after Leonhard Euler and Colin Maclaurin, who independently developed it in the 18th century. The Euler-Maclaurin formula is a summation formula that relates the sum of a function at discrete points to the integral of the function over a continuous interval.

Derivation of the Euler-Maclaurin Formula

To derive the Euler-Maclaurin formula, we start with the definition of a definite integral:

∫abf(x)dx=lim⁑nβ†’βˆžβˆ‘i=1nf(xi)Ξ”x\int_{a}^{b} f(x) dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x

where xix_i is a point in the interval [a,b][a,b] and Ξ”x\Delta x is the width of the interval.

We can rewrite the sum as:

βˆ‘i=1nf(xi)Ξ”x=βˆ‘i=1nf(xi)(bβˆ’an)\sum_{i=1}^{n} f(x_i) \Delta x = \sum_{i=1}^{n} f(x_i) \left( \frac{b-a}{n} \right)

Using the Taylor series expansion of f(xi)f(x_i) around xiβˆ’1x_{i-1}, we get:

f(xi)=f(xiβˆ’1)+fβ€²(xiβˆ’1)1!(xiβˆ’xiβˆ’1)+fβ€²β€²(xiβˆ’1)2!(xiβˆ’xiβˆ’1)2+β‹―f(x_i) = f(x_{i-1}) + \frac{f'(x_{i-1})}{1!} (x_i - x_{i-1}) + \frac{f''(x_{i-1})}{2!} (x_i - x_{i-1})^2 + \cdots

Substituting this into the sum, we get:

βˆ‘i=1nf(xi)(bβˆ’an)=βˆ‘i=1n(f(xiβˆ’1)+fβ€²(xiβˆ’1)1!(xiβˆ’xiβˆ’1)+fβ€²β€²(xiβˆ’1)2!(xiβˆ’xiβˆ’1)2+⋯ )(bβˆ’an)\sum_{i=1}^{n} f(x_i) \left( \frac{b-a}{n} \right) = \sum_{i=1}^{n} \left( f(x_{i-1}) + \frac{f'(x_{i-1})}{1!} (x_i - x_{i-1}) + \frac{f''(x_{i-1})}{2!} (x_i - x_{i-1})^2 + \cdots \right) \left( \frac{b-a}{n} \right)

Simplifying this expression, we get:

βˆ‘i=1nf(xi)(bβˆ’an)=βˆ‘i=1nf(xiβˆ’1)(bβˆ’an)+fβ€²(xiβˆ’1)1!(bβˆ’an)βˆ‘i=1n(xiβˆ’xiβˆ’1)+fβ€²β€²(xiβˆ’1)2!(bβˆ’an)2βˆ‘i=1n(xiβˆ’xiβˆ’1)2+β‹―\sum_{i=1}^{n} f(x_i) \left( \frac{b-a}{n} \right) = \sum_{i=1}^{n} f(x_{i-1}) \left( \frac{b-a}{n} \right) + \frac{f'(x_{i-1})}{1!} \left( \frac{b-a}{n} \right) \sum_{i=1}^{n} (x_i - x_{i-1}) + \frac{f''(x_{i-1})}{2!} \left( \frac{b-a}{n} \right)^2 \sum_{i=1}^{n} (x_i - x_{i-1})^2 + \cdots

Using the fact that βˆ‘i=1n(xiβˆ’xiβˆ’1)=bβˆ’a\sum_{i=1}^{n} (x_i - x_{i-1}) = b-a and βˆ‘i=1n(xiβˆ’xiβˆ’1)2=(bβˆ’a)22\sum_{i=1}^{n} (x_i - x_{i-1})^2 = \frac{(b-a)^2}{2}, we get:

βˆ‘i=1nf(xi)(bβˆ’an)=βˆ‘i=1nf(xiβˆ’1)(bβˆ’an)+fβ€²(xiβˆ’1)1!(bβˆ’an)(bβˆ’a)+fβ€²β€²(xiβˆ’1)2!(bβˆ’an)2(bβˆ’a)22+β‹―\sum_{i=1}^{n} f(x_i) \left( \frac{b-a}{n} \right) = \sum_{i=1}^{n} f(x_{i-1}) \left( \frac{b-a}{n} \right) + \frac{f'(x_{i-1})}{1!} \left( \frac{b-a}{n} \right) (b-a) + \frac{f''(x_{i-1})}{2!} \left( \frac{b-a}{n} \right)^2 \frac{(b-a)^2}{2} + \cdots

Taking the limit as nβ†’βˆžn \to \infty, we get:

∫abf(x)dx=βˆ‘i=1∞f(xiβˆ’1)(bβˆ’an)+fβ€²(xiβˆ’1)1!(bβˆ’an)(bβˆ’a)+fβ€²β€²(xiβˆ’1)2!(bβˆ’an)2(bβˆ’a)22+β‹―\int_{a}^{b} f(x) dx = \sum_{i=1}^{\infty} f(x_{i-1}) \left( \frac{b-a}{n} \right) + \frac{f'(x_{i-1})}{1!} \left( \frac{b-a}{n} \right) (b-a) + \frac{f''(x_{i-1})}{2!} \left( \frac{b-a}{n} \right)^2 \frac{(b-a)^2}{2} + \cdots

This is the Euler-Maclaurin formula as an asymptotic series.

Properties of the Euler-Maclaurin Formula

The Euler-Maclaurin formula has several important properties:

  • Convergence: The Euler-Maclaurin formula converges to the value of the definite integral as the number of terms increases.
  • Accuracy: The Euler-Maclaurin formula is more accurate than the trapezoidal rule for approximating the value of a definite integral.
  • Flexibility: The Euler-Maclaurin formula can be used to approximate the value of a definite integral with any degree of accuracy.

Applications of the Euler-Maclaurin Formula

The Euler-Maclaurin formula has several important applications:

  • Numerical integration: The Euler-Maclaurin formula is used to approximate the value of a definite integral in numerical integration.
  • Approximation of functions: The Euler-Maclaurin formula is used to approximate the value of a function at discrete points.
  • Analysis of algorithms: The Euler-Maclaurin formula is used to analyze the complexity of algorithms.

Conclusion

The Euler-Maclaurin formula is a powerful mathematical tool used to approximate the value of a definite integral. It is a summation formula that relates the sum of a function at discrete points to the integral of the function over a continuous interval. The Euler-Maclaurin formula has several important properties, including convergence, accuracy, and flexibility. It has several important applications, including numerical integration, approximation of functions, and analysis of algorithms.

References

  • Euler, L. (1738). "Methodus inveniendi lineas curvas maximi minimive proprietate gaudeant". Lausanne: Bousquet.
  • Maclaurin, C. (1742). "A Treatise of Algebra". London: Society for Promoting Christian Knowledge.
  • Abramowitz, M., & Stegun, I. A. (1964). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables". New York: Dover Publications.
    Euler-Maclaurin Formula as an Asymptotic Series: Q&A =====================================================

In this article, we will continue to explore the Euler-Maclaurin formula as an asymptotic series, answering some of the most frequently asked questions about this powerful mathematical tool.

Q: What is the Euler-Maclaurin formula?

A: The Euler-Maclaurin formula is a summation formula that relates the sum of a function at discrete points to the integral of the function over a continuous interval. It is a generalization of the trapezoidal rule, which is a method for approximating the value of a definite integral.

Q: How is the Euler-Maclaurin formula derived?

A: The Euler-Maclaurin formula is derived by using the Taylor series expansion of a function around a point. The Taylor series expansion is used to approximate the value of the function at a point, and then the sum of the approximations is used to relate the sum of the function at discrete points to the integral of the function over a continuous interval.

Q: What are the properties of the Euler-Maclaurin formula?

A: The Euler-Maclaurin formula has several important properties, including:

  • Convergence: The Euler-Maclaurin formula converges to the value of the definite integral as the number of terms increases.
  • Accuracy: The Euler-Maclaurin formula is more accurate than the trapezoidal rule for approximating the value of a definite integral.
  • Flexibility: The Euler-Maclaurin formula can be used to approximate the value of a definite integral with any degree of accuracy.

Q: What are the applications of the Euler-Maclaurin formula?

A: The Euler-Maclaurin formula has several important applications, including:

  • Numerical integration: The Euler-Maclaurin formula is used to approximate the value of a definite integral in numerical integration.
  • Approximation of functions: The Euler-Maclaurin formula is used to approximate the value of a function at discrete points.
  • Analysis of algorithms: The Euler-Maclaurin formula is used to analyze the complexity of algorithms.

Q: How is the Euler-Maclaurin formula used in numerical integration?

A: The Euler-Maclaurin formula is used in numerical integration to approximate the value of a definite integral. The formula is used to relate the sum of a function at discrete points to the integral of the function over a continuous interval. The sum of the function at discrete points is approximated using the Taylor series expansion, and then the Euler-Maclaurin formula is used to relate the sum to the integral.

Q: What are the advantages of using the Euler-Maclaurin formula?

A: The Euler-Maclaurin formula has several advantages, including:

  • High accuracy: The Euler-Maclaurin formula is more accurate than the trapezoidal rule for approximating the value of a definite integral.
  • Flexibility: The Euler-Maclaurin formula can be used to approximate the value of a definite integral with any degree of accuracy.
  • Convergence: The Euler-Maclaurin formula converges to the value of the definite integral as the number of terms increases.

Q: What are the disadvantages of using the Euler-Maclaurin formula?

A: The Euler-Maclaurin formula has several disadvantages, including:

  • Complexity: The Euler-Maclaurin formula is a complex formula that requires a good understanding of mathematical concepts.
  • Computational cost: The Euler-Maclaurin formula can be computationally expensive to implement.
  • Limited applicability: The Euler-Maclaurin formula is limited to approximating the value of definite integrals.

Conclusion

The Euler-Maclaurin formula is a powerful mathematical tool used to approximate the value of a definite integral. It is a summation formula that relates the sum of a function at discrete points to the integral of the function over a continuous interval. The Euler-Maclaurin formula has several important properties, including convergence, accuracy, and flexibility. It has several important applications, including numerical integration, approximation of functions, and analysis of algorithms.

References

  • Euler, L. (1738). "Methodus inveniendi lineas curvas maximi minimive proprietate gaudeant". Lausanne: Bousquet.
  • Maclaurin, C. (1742). "A Treatise of Algebra". London: Society for Promoting Christian Knowledge.
  • Abramowitz, M., & Stegun, I. A. (1964). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables". New York: Dover Publications.