{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
b. Expand sin t² and cos t² in a Maclaurin series, and then integrate to find the first four nonzero terms of the Maclaurin series for S and C.

Verified step by step guidance

1. Recall the Maclaurin series expansions for sine and cosine functions: \( \sin z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n+1}}{(2n+1)!} \) and \( \cos z = \sum_{n=0}^\infty (-1)^n \frac{z^{2n}}{(2n)!} \).
2. Substitute \( z = t^2 \) into these series to get the expansions for \( \sin t^2 \) and \( \cos t^2 \): \[ \sin t^2 = \sum_{n=0}^\infty (-1)^n \frac{t^{4n+2}}{(2n+1)!} \quad \text{and} \quad \cos t^2 = \sum_{n=0}^\infty (-1)^n \frac{t^{4n}}{(2n)!} \].
3. Write out the first four nonzero terms explicitly for both series: For \( \sin t^2 \): \[ t^2 - \frac{t^6}{3!} + \frac{t^{10}}{5!} - \frac{t^{14}}{7!} + \cdots \] For \( \cos t^2 \): \[ 1 - \frac{t^4}{2!} + \frac{t^8}{4!} - \frac{t^{12}}{6!} + \cdots \].
4. Integrate each term of the series for \( \sin t^2 \) from 0 to \( x \) to find the Maclaurin series for \( S(x) = \int_0^x \sin t^2 \, dt \): \[ \int_0^x t^{m} dt = \frac{x^{m+1}}{m+1} \], so integrate term-by-term accordingly.
5. Similarly, integrate each term of the series for \( \cos t^2 \) from 0 to \( x \) to find the Maclaurin series for \( C(x) = \int_0^x \cos t^2 \, dt \), again using term-by-term integration and the power rule for integrals.

Key Concepts

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, describing diffraction patterns and wave propagation.

Maclaurin Series Expansion

A Maclaurin series is a Taylor series expansion of a function about zero. It expresses functions as infinite sums of powers of the variable, with coefficients derived from derivatives at zero. Expanding sin(t²) and cos(t²) into Maclaurin series involves substituting t² into the known series for sine and cosine.

Term-by-Term Integration of Power Series

Once sin(t²) and cos(t²) are expanded into power series, integrating term-by-term from 0 to x is valid under uniform convergence. This process yields the Maclaurin series for S(x) and C(x) by integrating each power of t individually, resulting in a series representation for the Fresnel integrals.