Airy Functions. Chapter Introduction

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1 Chater 4 Airy Functions 4. Introduction Airy functions are named after the English astronomer George Biddell Airy (8 892). Airy s first mathematical work was on the diffraction henomenon, namely, the Airy disk the image of a oint object by a telescoe which is familiar to all of us in otics. The name Airy is connected with many hysical henomena and includes, besides the Airy disk, the Airy siral, an otical henomenon visible on quartz crystals, and the Airy stress function in elasticity. Airy was very interested in otics and in fact studied the formation of rainbows. A good qualitative summary of the rainbow is given by Adam. 4 In this aer Adam shows how the otical rainbow can be studied at many levels: (i) geometrical otics (rays), (ii) the Airy aroximation, (iii) Mie scattering, (iv) comlex angular momentum, and (v) catastrohe theory. Airy s analysis is aroximate but alies well to large raindros that make u the common rainbow (for small dros, catastrohe theory has been used). Details of Airy s analysis can be found in the chater on the otics of raindros in the book by van de Hulst 4 (see also Berry 42 ). Airy also analyzed the intensity of light near a caustic wavefront. During his investigation utilizing the scalar diffraction integral, he introduced a function W(m) defined by the integral Z` WðmÞ ¼ cos 2 ðv mvþ dv (4.) as a solution of the differential equation d 2 W dv 2 þ 2 mw ¼ : (4.2) 2 Jeffreys 4 introduced the modern notation currently used: AiðxÞ ¼ Z` t cos þ xt dt: (4.) 7

2 72 Chater 4 Equation (4.) is referred to as the Airy integral and can be shown to be the solution to a homogeneous differential equation of the tye d 2 y ¼ xy: (4.4) 2 dx This equation is generally known as the Airy equation or the Airy differential equation. However, caution must be exercised in differentiating Eq. (4.) under the integral, since the integral would become indeterminate as t! `. For a rigorous roof of this solution, we need to use comlex variable techniques. Here, we show a simle intuitive technique. Let s define a function F s (x) as follows: F s ðxþ ¼ Z s cos tx þ t dt, (4.5) where s is a large but finite number. Substituting this F s for y in Eq. (4.4), we obtain d 2 F s dx 2 xf sðxþ ¼ Z s ðt 2 þ xþ cos tx dt t ¼ sx sin þ s : (4.6) As s! `, the function oscillates raidly between and þ. Therefore, we can set the mean value of the function equal to zero and show that F s is a solution of Eq. (4.4) in the limit s! `. In this limit F s becomes the Airy integral [Eq. (4.)]. 4.2 Ai(x) and Bi(x) Functions Ai(x) can be given by a ower series exansion AiðxÞ ¼ 2 Gð 2 Þ Gð Þ þ! x þ ðþð4þ 6 x þ 2 4! x4 þ ð2þð5þ 7! x 6 þ ðþð4þð7þ x 9 9! x 7 þ ð2þð5þð8þ x! : (4.7) The Airy differential equation [Eq. (4.4)] is a second-order differential equation; it must, therefore, have a second indeendent solution. This is denoted as Bi(x) and is given by BiðxÞ ¼ Z` ex t þ xt þ sin t þ xt dt: (4.8)

3 Airy Functions 7 Figure 4. The Airy functions Ai(x) (solid line) and Bi(x) (dotted line). The ower series exansion of Bi(x) is given as ffiffiffi BiðxÞ ¼ 2 Gð 2 ffiffiffi Þ þ Gð Þ þ! x þ ðþð4þ x 6 þ 6! x þ 2 4! x4 þ ð2þð5þ x 7 þ 7! : (4.9) Functions Ai(x) and Bi(x) are the Airy functions. These functions are available as airy in sciy.secial in Python. This function returns four arrays, Ai, Ai, Bi, and Bi in that order. Figure 4. shows the lots of Airy functions Ai and Bi. As is usual, let us write a ower series solution of the form yðxþ ¼a þ a x þ a 2 x 2 þ (4.) to solve the Airy equation. Substituting Eq. (4.) in Eq. (4.4) and simlifying, we obtain 2a 2 þðþð2þa x þð4þðþa 4 x 2 þ þnðn Þa n x n 2 þ ¼ a x þ a x 2 þ a 2 x þ þa n x n 2 þ : (4.)

4 74 Chater 4 Equating coefficients, we find that a 2 ¼, and consequently, we obtain for the solution, a ¼ a ð2þðþ, a 4 ¼ a ðþð4þ. a n ¼ a n nðn Þ, (4.2) yðxþ ¼a þ x2 ð2þðþ þ x 6 ð2þðþð5þð6þ þ þ a x þ x4 ðþð4þ þ x 7 ðþð4þð6þð7þ þ, (4.) where a and a are the two arbitrary constants that need to be fixed alying the aroriate boundary conditions. If we were to write f ðxþ ¼ þ! x þ ðþð4þ 6! x 6 þ ðþð4þð7þ x 9 þ 9! (4.4a) gðxþ ¼x þ 2 4! x4 þ ð2þð5þ 7! x 7 þ ð2þð5þð8þ x þ,! we notice that Eq. (4.) could be written as y ¼ a þ! x þ ðþð4þ x 6 þ ðþð4þð7þ x 9 þ 6! 9! þ a x þ 2 4! x4 þ ð2þð5þ x 7 þ ð2þð5þð8þ x þ 7!! If we set a ¼ ¼.55 2 G 2 (4.4b) : (4.5) (4.6a) a ¼ ¼.25882, G (4.6b)

5 Airy Functions 75 we obtain the series exansion Ai(x) [Eq. (4.7)] and Bi(x) [Eq. (4.9)] as AiðxÞ ¼a f ðxþ a gðxþ (4.7a) BiðxÞ ¼ ffiffiffi ½a f ðxþþa gðxþš: (4.7b) Equations (4.7a) and (4.7b) are the ways these two functions Ai(x) and Bi(x) are traditionally written. It is easy to show, from Eqs. (4.5) and (4.7), that AiðÞ ¼a, Ai ðþ ¼ a, BiðÞ ¼ ffiffiffi a, Bi ðþ ¼ ffiffiffi a, (4.8) where the rime ( ) is used to denote the derivative, and f () is a short form for df dx j x¼. For higher derivatives we can show that ðn þ Þ Ai ðnþ ðþ ¼ð Þ n c n sin (4.9a) ð4n þ Þ Bi ðnþ ðþ ¼c n þ sin (4.9b) 6 with C n ¼ n þ ðn 2Þ G : (4.2) Here, the suerscrit n denotes the n-th derivative. We can get the ascending series of the derivatives by differentiating f(x) and g(x) in Eq. (4.7) term by term, as shown below: Ai ðxþ¼a f ðxþ a g ðxþ Bi ðxþ¼ ffiffi ½a f ðxþþa g ðxþš f ðxþ¼ x2 2 þ x 5 ð2þðþ 5 þ x 8 ð2þðþð5þð6þ 8 þ g ðxþ¼þ x ðþðþ þ x 6 ðþðþð4þð6þ 6 þ x 9 þ : (4.2) ðþðþð4þð6þð7þð9þ 9 4. Relationshi with Bessel Functions In Sec. 2.2, we mentioned that the beta function could be written in terms of the Bessel function. Similarly, through Eqs. (.77), we related the Fresnel integral to the Bessel functions. In a similar way, we deal with the Bessel function before it makes its aearance in this book (see Ch. 5). This is because Ai(x) and Bi(x) can be exressed in terms of the Bessel function, and

6 76 Chater 4 using the asymtotic forms of the Bessel function, it is ossible to get the asymtotic form of the Airy function. Readers may choose to ski this section and come back to it after working through Ch. 5. Consider Eq. (4.4), d 2 yðxþ dx 2 xyðxþ ¼ (4.22) in the region x,. Let us define z ¼ x. Therefore, for z., we obtain d 2 yðzþ dz 2 þ zyðzþ ¼: (4.2) Let y(z) ¼ z /2 f(z). The above equation, therefore, becomes z 2 d2 f df þ z 2 2 dz dz þ 4 z 2 þ z 2 fðzþ ¼: (4.24) Now, we make another transformation of the variable z ¼ 2 z 2. This results in the following transformations of the derivatives: z ¼ 2 z 2 df dz ¼ df dz dz dz ¼ z 2 df dz d 2 f dz 2 ¼ d df dz z 2 dz dz dz þ 2 z 2 df dz ¼ z d2 f dz 2 þ df z 2 2 dz : (4.25a) (4.25b) (4.25c) Substituting these exressions for the derivatives into Eq. (4.24) and simlifying, we end u with z 2 d2 f dz 2 þ df 2 dz þ 4 z 2 þ z 2 f ¼ : (4.26) Using the transformation z ¼ 2 z 2, we can simlify the above equation to obtain z 2 d2 f dz þ z df dz þ z 2 f ¼ : (4.27) 9 As will be seen in Ch. 5 [Eq. (5.)], the solution of the differential equation x 2 d2 y dx 2 þ x dy dx þðx2 n 2 Þy ¼ (4.28)

7 Airy Functions 77 is the Bessel function of the order n. Therefore, Eq. (4.27) reresents a Bessel differential equation of the order. Since it is a second-order differential equation, it has two solutions, namely Bessel functions of the order. The two indeendent solutions of the equation are and y ¼jxj 2 J ðzþ, y ¼jxj 2 J ðzþ, where z ¼ 2 z 2 ¼ 2 jxj 2. The aroriate linear combinations of the Airy functions for x, are, therefore, A ð xþ ¼ ffiffiffi x ½J ðzþþj þ ðzþš, (4.29a) rffiffiffi x and B ð xþ ¼ ½J ðzþ J þ ðzþš, (4.29b) with the understanding that z ¼ 2 jxj 2. We can do a similar analysis for the region x. and obtain as two indeendent solutions, and y ¼ x 2 I z, y ¼ x 2 I ðzþ, where I n reresents the modified Bessel function (see Sec. 5.6). As above, the Airy functions then become ffiffiffi x AiðxÞ ¼ ½I ðzþ I þ ðzþš (4.a) rffiffiffi x BiðxÞ ¼ ½I ðzþþi þ ðzþš: (4.b) Airy functions are thus Bessel functions or linear combinations of these functions of the order. Jeffreys44 makes an interesting observation about this relationshi between the Bessel functions and the Airy functions: Bessel functions of order seem to have no alication excet to rovide an inconvenient way of exressing this [Airy] function.

Sequences, Series, and the Binomial Formula

Sequences, Series, and the Binomial Formula CHAPTER Sequences, Series, nd the Binomil Formul. SEQUENCES. ; ; ; ; 6 ; 6 6. ðþ ; ðþ ; ð Þ 6; ðþ ; ðþ 6; 6 ð6þ. ðþ ; ðþ : ðþ ; ðþ ; ðþ ; 6 ðþ 6 6 6. ; ; ; ; ; 6 6 6. 0 ; ; ; 8 ; 6 8 ; 6. 0; ; 6 ; ; 6