Fresnel integral – Wikipedia

2022-07-29 21:29:34

Particular operate outlined by an integral

Plots of

S (x)

and

C (x)

. The utmost of

C (x)

is about 0.977451424. If the integrands of

S

and

C

have been outlined utilizing

π / 2 t 2

as a substitute of

t 2

, then the picture could be scaled vertically and horizontally (see under).

The Fresnel integrals $S (x)$ and $C (x)$ are two transcendental functions named after Augustin-Jean Fresnel which are utilized in optics and are carefully associated to the error function ( $erf$ ). They come up within the description of near-field Fresnel diffraction phenomena and are outlined by the next integral representations:

{displaystyle S(x)=int _{0}^{x}sin left(t^{2}proper),dt,quad C(x)=int _{0}^{x}cos left(t^{2}proper),dt.}

The simultaneous parametric plot of $S (x)$ and $C (x)$ is the Euler spiral (often known as the Cornu spiral or clothoid).

Definition[edit]

Fresnel integrals with arguments

π / 2 t 2

as a substitute of

t 2

converge to 12 as a substitute of

1 / 2 \cdot \sqrt π / 2

The Fresnel integrals admit the next power series expansions that converge for all $x$ :

{displaystyle {start{aligned}S(x)&=int _{0}^{x}sin left(t^{2}proper),dt=sum _{n=0}^{infty }(-1)^{n}{frac {x^{4n+3}}{(2n+1)!(4n+3)}},C(x)&=int _{0}^{x}cos left(t^{2}proper),dt=sum _{n=0}^{infty }(-1)^{n}{frac {x^{4n+1}}{(2n)!(4n+1)}}.finish{aligned}}}

Some broadly used tables use $π / 2 t 2$ as a substitute of $t 2$ for the argument of the integrals defining $S (x)$ and $C (x)$ . This adjustments their limits at infinity from $1 / 2 \cdot \sqrt π / 2$ to 1/2 and the arc size for the primary spiral flip from $\sqrt 2 π$ to 2 (at $t = 2$ ). These different capabilities are often generally known as normalized Fresnel integrals.

Euler spiral[edit]

Euler spiral

(x, y) = (C (t), S (t))

. The spiral converges to the centre of the holes within the picture as

t

tends to optimistic or destructive infinity.

Animation depicting evolution of a Cornu spiral with the tangential circle with the identical radius of curvature as at its tip, often known as an osculating circle.

The Euler spiral, often known as Cornu spiral or clothoid, is the curve generated by a parametric plot of $S (t)$ in opposition to $C (t)$ . The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.

From the definitions of Fresnel integrals, the infinitesimals $dx$ and $dy$ are thus:

{displaystyle {start{aligned}dx&=C'(t),dt=cos left(t^{2}proper),dt,dy&=S'(t),dt=sin left(t^{2}proper),dt.finish{aligned}}}

Thus the size of the spiral measured from the origin could be expressed as

{displaystyle L=int _{0}^{t_{0}}{sqrt {dx^{2}+dy^{2}}}=int _{0}^{t_{0}}dt=t_{0}.}

That’s, the parameter $t$ is the curve size measured from the origin $(0, 0)$ , and the Euler spiral has infinite size. The vector $(cos(t 2), sin(t 2))$ additionally expresses the unit tangent vector alongside the spiral, giving $θ = t 2$ . Since $t$ is the curve size, the curvature $κ$ could be expressed as

{displaystyle kappa ={frac {1}{R}}={frac {dtheta }{dt}}=2t.}

Thus the speed of change of curvature with respect to the curve size is

{displaystyle {frac {dkappa }{dt}}={frac {d^{2}theta }{dt^{2}}}=2.}

An Euler spiral has the property that its curvature at any level is proportional to the space alongside the spiral, measured from the origin. This property makes it helpful as a transition curve in freeway and railway engineering: if a car follows the spiral at unit velocity, the parameter $t$ within the above derivatives additionally represents the time. Consequently, a car following the spiral at fixed velocity may have a continuing charge of angular acceleration.

Sections from Euler spirals are generally integrated into the form of rollercoaster loops to make what are generally known as clothoid loops.

Properties[edit]

$C (x)$ and $S (x)$ are odd functions of $x$ ,

{displaystyle C(-x)=-C(x),quad S(-x)=-S(x).}

Asymptotics of the Fresnel integrals as $x \to \infty$ are given by the formulation:

{displaystyle {start{aligned}S(x)&={sqrt {{tfrac {1}{8}}pi }}operatorname {sgn} x-left[1+Oleft(x^{-4}right)right]left({frac {cos left(x^{2}proper)}{2x}}+{frac {sin left(x^{2}proper)}{4x^{3}}}proper),[6px]C(x)&={sqrt {{tfrac {1}{8}}pi }}operatorname {sgn} x+left[1+Oleft(x^{-4}right)right]left({frac {sin left(x^{2}proper)}{2x}}-{frac {cos left(x^{2}proper)}{4x^{3}}}proper).finish{aligned}}}

Complicated Fresnel integral

S (z)

Utilizing the facility sequence expansions above, the Fresnel integrals could be prolonged to the area of complex numbers, the place they turn into analytic functions of a fancy variable.

$C (z)$ and $S (z)$ are entire functions of the complicated variable $z$ .

The Fresnel integrals could be expressed utilizing the error function as follows:^[4]

Complicated Fresnel integral

C (z)

{displaystyle {start{aligned}S(z)&={sqrt {frac {pi }{2}}}cdot {frac {1+i}{4}}left[operatorname {erf} left({frac {1+i}{sqrt {2}}}zright)-ioperatorname {erf} left({frac {1-i}{sqrt {2}}}zright)right],[6px]C(z)&={sqrt {frac {pi }{2}}}cdot {frac {1-i}{4}}left[operatorname {erf} left({frac {1+i}{sqrt {2}}}zright)+ioperatorname {erf} left({frac {1-i}{sqrt {2}}}zright)right].finish{aligned}}}

{displaystyle {start{aligned}C(z)+iS(z)&={sqrt {frac {pi }{2}}}cdot {frac {1+i}{2}}operatorname {erf} left({frac {1-i}{sqrt {2}}}zright),[6px]S(z)+iC(z)&={sqrt {frac {pi }{2}}}cdot {frac {1+i}{2}}operatorname {erf} left({frac {1+i}{sqrt {2}}}zright).finish{aligned}}}

Limits as $x$ approaches infinity[edit]

The integrals defining $C (x)$ and $S (x)$ can’t be evaluated within the closed form by way of elementary functions, besides in particular circumstances. The limits of those capabilities as $x$ goes to infinity are identified:

{displaystyle int _{0}^{infty }cos left(t^{2}proper),dt=int _{0}^{infty }sin left(t^{2}proper),dt={frac {sqrt {2pi }}{4}}={sqrt {frac {pi }{8}}}approx 0.6267.}

The sector contour used to calculate the boundaries of the Fresnel integrals

This may be derived with any one among a number of strategies. One in every of them^[5] makes use of a contour integral of the operate

{displaystyle e^{-z^{2}}}

across the boundary of the sector-shaped area within the complex plane fashioned by the optimistic $x$ -axis, the bisector of the primary quadrant $y = x$ with $x \geq 0$ , and a round arc of radius $R$ centered on the origin.

As $R$ goes to infinity, the integral alongside the round arc $γ 2$ tends to $0$

{displaystyle left|int _{gamma _{2}}e^{-z^{2}},dzright|=left|int _{0}^{frac {pi }{4}}e^{-R^{2}(cos t+isin t)^{2}},Re^{it}dtright|leq Rint _{0}^{frac {pi }{4}}e^{-R^{2}cos 2t},dtleq Rint _{0}^{frac {pi }{4}}e^{-R^{2}left(1-{frac {4}{pi }}tright)},dt={frac {pi }{4R}}left(1-e^{-R^{2}}proper),}

the place polar coordinates $z = Re it$ have been used and Jordan’s inequality was utilised for the second inequality. The integral alongside the actual axis $γ 1$ tends to the half Gaussian integral

{displaystyle int _{gamma _{1}}e^{-z^{2}},dz=int _{0}^{infty }e^{-t^{2}},dt={frac {sqrt {pi }}{2}}.}

Be aware too that as a result of the integrand is an entire function on the complicated airplane, its integral alongside the entire contour is zero. General, we should have

{displaystyle int _{gamma _{3}}e^{-z^{2}},dz=int _{gamma _{1}}e^{-z^{2}},dz=int _{0}^{infty }e^{-t^{2}},dt,}

the place $γ 3$ denotes the bisector of the primary quadrant, as within the diagram. To guage the left hand aspect, parametrize the bisector as

{displaystyle z=te^{i{frac {pi }{4}}}={frac {sqrt {2}}{2}}(1+i)t}

the place $t$ ranges from 0 to $+\infty$ . Be aware that the sq. of this expression is simply $+ it 2$ . Due to this fact, substitution offers the left hand aspect as

{displaystyle int _{0}^{infty }e^{-it^{2}}{frac {sqrt {2}}{2}}(1+i),dt.}

Utilizing Euler’s formula to take actual and imaginary elements of $e - it 2$ offers this as

{displaystyle {start{aligned}&int _{0}^{infty }left(cos left(t^{2}proper)-isin left(t^{2}proper)proper){frac {sqrt {2}}{2}}(1+i),dt[6px]&quad ={frac {sqrt {2}}{2}}int _{0}^{infty }left[cos left(t^{2}right)+sin left(t^{2}right)+ileft(cos left(t^{2}right)-sin left(t^{2}right)right)right],dt[6px]&quad ={frac {sqrt {pi }}{2}}+0i,finish{aligned}}}

the place we’ve written $0 i$ to emphasise that the unique Gaussian integral’s worth is totally actual with zero imaginary half. Letting

{displaystyle I_{C}=int _{0}^{infty }cos left(t^{2}proper),dt,quad I_{S}=int _{0}^{infty }sin left(t^{2}proper),dt}

after which equating actual and imaginary elements produces the next system of two equations within the two unknowns $I C$ and $I S$ :

{displaystyle {start{aligned}I_{C}+I_{S}&={sqrt {frac {pi }{2}}},I_{C}-I_{S}&=0.finish{aligned}}}

Fixing this for $I C$ and $I S$ offers the specified end result.

Generalization[edit]

The integral

{displaystyle int x^{m}e^{ix^{n}},dx=int sum _{l=0}^{infty }{frac {i^{l}x^{m+nl}}{l!}},dx=sum _{l=0}^{infty }{frac {i^{l}}{(m+nl+1)}}{frac {x^{m+nl+1}}{l!}}}

is a confluent hypergeometric function and in addition an incomplete gamma function

{displaystyle {start{aligned}int x^{m}e^{ix^{n}},dx&={frac {x^{m+1}}{m+1}},_{1}F_{1}left({start{array}{c}{frac {m+1}{n}}1+{frac {m+1}{n}}finish{array}}mid ix^{n}proper)[6px]&={frac {1}{n}}i^{frac {m+1}{n}}gamma left({frac {m+1}{n}},-ix^{n}proper),finish{aligned}}}

which reduces to Fresnel integrals if actual or imaginary elements are taken:

{displaystyle int x^{m}sin(x^{n}),dx={frac {x^{m+n+1}}{m+n+1}},_{1}F_{2}left({start{array}{c}{frac {1}{2}}+{frac {m+1}{2n}}{frac {3}{2}}+{frac {m+1}{2n}},{frac {3}{2}}finish{array}}mid -{frac {x^{2n}}{4}}proper).}

The main time period within the asymptotic growth is

{displaystyle _{1}F_{1}left({start{array}{c}{frac {m+1}{n}}1+{frac {m+1}{n}}finish{array}}mid ix^{n}proper)sim {frac {m+1}{n}},Gamma left({frac {m+1}{n}}proper)e^{ipi {frac {m+1}{2n}}}x^{-m-1},}

and subsequently

{displaystyle int _{0}^{infty }x^{m}e^{ix^{n}},dx={frac {1}{n}},Gamma left({frac {m+1}{n}}proper)e^{ipi {frac {m+1}{2n}}}.}

For $m = 0$ , the imaginary a part of this equation particularly is

{displaystyle int _{0}^{infty }sin left(x^{a}proper),dx=Gamma left(1+{frac {1}{a}}proper)sin left({frac {pi }{2a}}proper),}

with the left-hand aspect converging for $a > 1$ and the right-hand aspect being its analytical extension to the entire airplane much less the place lie the poles of $Γ (a -1)$ .

The Kummer transformation of the confluent hypergeometric operate is

{displaystyle int x^{m}e^{ix^{n}},dx=V_{n,m}(x)e^{ix^{n}},}

with

{displaystyle V_{n,m}:={frac {x^{m+1}}{m+1}},_{1}F_{1}left({start{array}{c}11+{frac {m+1}{n}}finish{array}}mid -ix^{n}proper).}

Numerical approximation[edit]

For computation to arbitrary precision, the facility sequence is appropriate for small argument. For big argument, asymptotic expansions converge quicker. Continued fraction strategies may be used.

For computation to specific goal precision, different approximations have been developed. Cody developed a set of environment friendly approximations based mostly on rational capabilities that give relative errors all the way down to 2×10⁻¹⁹. A FORTRAN implementation of the Cody approximation that features the values of the coefficients wanted for implementation in different languages was printed by van Snyder. Boersma developed an approximation with error lower than 1.6×10⁻⁹.

Functions[edit]

The Fresnel integrals have been initially used within the calculation of the electromagnetic area depth in an atmosphere the place mild bends round opaque objects. Extra lately, they’ve been used within the design of highways and railways, particularly their curvature transition zones, see track transition curve. Different purposes are rollercoasters or calculating the transitions on a velodrome observe to permit fast entry to the bends and gradual exit.^{[citation needed]}

See additionally[edit]

References[edit]

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Alazah, Mohammad (2012). “Computing Fresnel integrals by way of modified trapezium guidelines”. Numerische Mathematik. 128 (4): 635–661. arXiv:1209.3451. Bibcode:2012arXiv1209.3451A. doi:10.1007/s00211-014-0627-z. S2CID 13934493.
Beatty, Thomas (2013). “How to evaluate Fresnel Integrals” (PDF). FGCU Math – Summer season 2013. Retrieved 27 July 2013.
Boersma, J. (1960). “Computation of Fresnel Integrals”. Math. Comp. 14 (72): 380. doi:10.1090/S0025-5718-1960-0121973-3. MR 0121973.
Bulirsch, Roland (1967). “Numerical calculation of the sine, cosine and Fresnel integrals”. Numer. Math. 9 (5): 380–385. doi:10.1007/BF02162153. S2CID 121794086.
Cody, William J. (1968). “Chebyshev approximations for the Fresnel integrals” (PDF). Math. Comp. 22 (102): 450–453. doi:10.1090/S0025-5718-68-99871-2.
Hangelbroek, R. J. (1967). “Numerical approximation of Fresnel integrals via Chebyshev polynomials”. J. Eng. Math. 1 (1): 37–50. Bibcode:1967JEnMa…1…37H. doi:10.1007/BF01793638. S2CID 122271446.
Mathar, R. J. (2012). “Sequence Enlargement of Generalized Fresnel Integrals”. arXiv:1211.3963 [math.CA].
Nave, R. (2002). “The Cornu spiral”. (Makes use of $π / 2 t 2$ as a substitute of $t 2$ .)
Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007). “Section 6.8.1. Fresnel Integrals”. Numerical Recipes: The Artwork of Scientific Computing (third ed.). New York: Cambridge College Press. ISBN 978-0-521-88068-8.
van Snyder, W. (1993). “Algorithm 723: Fresnel integrals”. ACM Trans. Math. Softw. 19 (4): 452–456. doi:10.1145/168173.168193. S2CID 12346795.
Stewart, James (2008). Calculus Early Transcendentals. Cengage Studying EMEA. ISBN 978-0-495-38273-7.
Temme, N. M. (2010), “Error Functions, Dawson’s and Fresnel Integrals”, in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge College Press, ISBN 978-0-521-19225-5, MR 2723248
van Wijngaarden, A.; Scheen, W. L. (1949). Desk of Fresnel Integrals. Verhandl. Konink. Ned. Akad. Wetenschapen. Vol. 19.
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