Fresnel Integral

The Fresnel integrals are a pair of transcendental functions that arise in the study of wave optics, particularly in problems involving diffraction and interference. These integrals are defined as follows:

$$S(x)={\int }_0^x\sin \left(\frac{\pi t^2}{2}\right)dt$$ $$C(x)={\int }_0^x\cos \left(\frac{\pi t^2}{2}\right)dt$$

They are often written together as a parametric curve in the complex plane:

$$Z(x)=C(x)+iS(x)$$

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Basic Properties

• Both $C(x)$ and $S(x)$ are odd functions: $C(-x)=-C(x)$ and $S(-x)=-S(x)$.
• As $x\to \infty$, both $C(x)$ and $S(x)$ tend toward $\frac{1}{2}$, leading to:

${\lim }_{x\to \infty }C(x)={\lim }_{x\to \infty }S(x)=\frac{1}{2}$

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Applications

The Fresnel integrals are widely used in physics and engineering, including:

• Wave optics: They describe the intensity distribution in a diffraction pattern (e.g., near-field diffraction).
• Signal processing: They help in analyzing signals involving chirps or oscillations.
• Electromagnetics: They model certain problems in electromagnetic wave propagation.

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Approximations

For small $x$, we can approximate $C(x)$ and $S(x)$ as:

$C(x)\approx x-\frac{\pi x^5}{40},S(x)\approx \frac{\pi x^3}{6}-\frac{\pi x^7}{336}$

These approximations become useful for practical numerical computations.

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Plot of Fresnel Integrals

The parametric plot of $C(x)$ and $S(x)$ forms the well-known Cornu spiral, a tool used in optical design and diffraction analysis.

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Trivia

The Fresnel integrals were introduced by the French engineer Augustin-Jean Fresnel in the early 19th century during his studies on light diffraction.

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Further Reading

• Fresnel Diffraction and the Cornu Spiral
• Applications in Signal Processing: Refer to standard texts on Fourier transforms and wave propagation.