{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integrals
S(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dt
e. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?

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Recall that the Fresnel integral \( C(x) = \int_0^x \cos(t^2) \, dt \) can be expanded into a Maclaurin series by expanding \( \cos(t^2) \) as a power series and integrating term-by-term.

Write the Maclaurin series for \( \cos(z) \), which is \( \cos(z) = \sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!} \). Substitute \( z = t^2 \) to get \( \cos(t^2) = \sum_{n=0}^\infty (-1)^n \frac{t^{4n}}{(2n)!} \).

Integrate the series term-by-term from 0 to \( x \) to find the series for \( C(x) \): \[ C(x) = \sum_{n=0}^\infty (-1)^n \frac{1}{(2n)!} \int_0^x t^{4n} \, dt = \sum_{n=0}^\infty (-1)^n \frac{x^{4n+1}}{(4n+1)(2n)!} \].

To approximate \( C(-0.25) \) with an error no greater than \( 10^{-6} \), use the alternating series remainder estimation. The error is less than or equal to the absolute value of the first omitted term in the series.

Find the smallest integer \( N \) such that the absolute value of the \( (N+1)^{th} \) term: \[ \left| (-1)^{N+1} \frac{(-0.25)^{4(N+1)+1}}{(4(N+1)+1)(2(N+1))!} \right| \leq 10^{-6} \]. This will tell you how many terms are needed to achieve the desired accuracy.

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주요 개념

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Fresnel Integrals

Fresnel integrals S(x) and C(x) are defined as integrals of sine and cosine functions with quadratic arguments, specifically S(x) = ∫₀ˣ sin(t²) dt and C(x) = ∫₀ˣ cos(t²) dt. They arise in optics and wave theory, representing diffraction patterns and are not expressible in elementary functions, often requiring series expansions or numerical methods for evaluation.

Maclaurin Series Expansion

A Maclaurin series is a Taylor series expansion of a function about zero, expressing it as an infinite sum of derivatives at zero multiplied by powers of x. For functions like cos(t²), the Maclaurin series allows approximation by polynomials, facilitating numerical evaluation of integrals like C(x) by integrating term-by-term.

Error Estimation in Series Approximation

When approximating functions with series, the error is the difference between the true value and the partial sum. To ensure the error is below a threshold (e.g., 10⁻⁶), one uses remainder estimates or bounds on the next term in the series. This guides how many terms are needed for a desired accuracy.

{Use of Tech} Fresnel integrals The theory of optics gives rise to the two Fresnel integralsS(x) = ∫₀ˣ sin t² dt and C(x) = ∫₀ˣ cos t² dte. How many terms of the Maclaurin series are required to approximate C(−0.25) with an error no greater than 10⁻⁶?