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Special Functions for Optical Science and Engineering [Pehme köide]

,
  • Formaat: Paperback / softback, 408 pages, kõrgus x laius: 229x152 mm, kaal: 721 g
  • Sari: Tutorial Texts
  • Ilmumisaeg: 01-Oct-2015
  • Kirjastus: SPIE Press
  • ISBN-10: 1628418877
  • ISBN-13: 9781628418873
Teised raamatud teemal:
Special Functions for Optical Science and EngineeringSuurem pilt
  • Formaat: Paperback / softback, 408 pages, kõrgus x laius: 229x152 mm, kaal: 721 g
  • Sari: Tutorial Texts
  • Ilmumisaeg: 01-Oct-2015
  • Kirjastus: SPIE Press
  • ISBN-10: 1628418877
  • ISBN-13: 9781628418873
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781628418873.html
Märksõnad:
This tutorial text is for those who use special functions in their work or study but are not mathematicians. Traditionally, special functions arise as solutions to certain linear second-order differential equations with variable coefficientsequations having applications in physics, chemistry, engineering, etc. This book introduces these differential equations, their solutions, and their applications in optical science and engineering. In addition to the common special functions, some less common functions are included. Also covered are Zernike polynomials, which are widely used in characterizing the quality of any imaging system, as well as certain integral transforms not usually covered in elementary texts.

The mathematical treatment is kept at a low level (knowledge of elementary trigonometric and exponential functions, simple calculus, and solution of differential equations using the separation of variables technique is required). The book is liberally illustrated, and almost every chapter includes a set of Python 3.x codes that illustrate the use of these functions. Readers with a modest introduction to programming concepts will be able to modify these sample codes as needed.
Preface xvii
Acknowledgment xix
1 Introduction 1(20)
1.1 Some Famous Equations
1(1)
1.2 Linearity
2(1)
1.3 Maxwell's Equations
2(2)
1.4 Curvilinear Coordinates
4(7)
1.4.1 Spherical polar coordinates
6(3)
1.4.2 Circular cylindrical coordinates
9(1)
1.4.3 Elliptical cylindrical coordinates
10(1)
1.5 Solution to the Helmholtz Equation
11(3)
1.6 What's So Special About Special Functions?
14(1)
1.7 Python Codes
14(7)
2 Gamma, Beta, and Error Functions 21(24)
2.1 Gamma Function
21(11)
2.1.1 Factorial function
22(1)
2.1.2 Stirling's approximation
23(2)
2.1.3 Gamma function at some special points
25(6)
2.1.3.1 Gamma at half integer values
25(3)
2.1.3.2 Gamma of negative numbers
28(1)
2.1.3.3 Legendre duplication formula
28(2)
2.1.3.4 Reflection formula
30(1)
2.1.4 The incomplete gamma function
31(1)
2.1.5 Digamma function
32(1)
2.2 Beta Function
32(5)
2.2.1 Properties of the beta function
33(2)
2.2.2 Applications of the beta function
35(2)
2.3 Error Function
37(8)
2.3.1 Error function and the normal distribution
38(1)
2.3.2 Error function and the gamma function
39(1)
2.3.3 Series expansion for the error function
39(1)
2.3.4 Properties of the error function
40(1)
2.3.5 Complementary error function
40(5)
3 Other Integral Functions 45(26)
3.1 Exponential Integrals
45(3)
3.2 Logarithmic, Sine, and Cosine Integrals
48(3)
3.3 Delta Function
51(6)
3.3.1 Properties of the delta function
53(3)
3.3.2 Application of the delta function
56(1)
3.4 Kronecker Delta
57(1)
3.5 Fresnel Integrals
58(8)
3.6 Elliptic Integrals
66(5)
4 Airy Functions 71(20)
4.1 Introduction
71(1)
4.2 Ai(x) and Bi(x) Functions
72(3)
4.3 Relationship with Bessel Functions
75(3)
4.4 Asymptotic Behavior of the Airy Function
78(2)
4.5 Some Applications
80(11)
4.5.1 Intensity near a caustic
80(2)
4.5.2 Airy beams
82(4)
4.5.3 Reflection of electromagnetic waves by the ionosphere
86(2)
4.5.4 Schrodinger equation
88(2)
4.5.5 Airy functions and the JWKB approximation
90(1)
5 Bessel Functions 91(42)
5.1 Introduction
91(1)
5.2 Bessel Function of the First Kind
92(5)
5.3 Bessel Functions of the Second Kind-Neumann Functions
97(2)
5.4 Generalized Bessel Differential Equation
99(1)
5.5 Bessel Functions of the Third Kind-Hankel Functions
100(2)
5.6 Modified Bessel Functions
102(2)
5.7 Kelvin Functions
104(1)
5.8 Spherical Bessel Functions
105(4)
5.9 Generating Function for Bessel Functions
109(3)
5.10 Integral Relationship of Bessel Functions
112(5)
5.11 Recurrence Relations of Bessel Functions
117(3)
5.12 Approximate Formulas
120(1)
5.13 Bessel-Fourier Series and Orthogonality
120(3)
5.14 Examples
123(6)
5.14.1 Modes of an optical waveguide
123(3)
5.14.2 The Kaiser-Bessel window
126(1)
5.14.3 Bessel beams
127(2)
5.15 A Note on Python Coding
129(2)
5.16 Summary
131(2)
6 Chebyshev Polynomials 133(12)
6.1 Introduction
133(1)
6.2 Definition of Tn(x)and Un(x)
133(6)
6.3 Recurrence Relations
139(1)
6.4 Special Values
140(1)
6.5 Generating Function
140(2)
6.6 Orthogonality
142(1)
6.7 Chebyshev Series
143(1)
6.8 Applications
144(1)
7 Hermite Polynomials 145(26)
7.1 Introduction
145(1)
7.2 Solution to the Hermite Equation
145(3)
7.3 Series Solution
148(2)
7.4 Generating Function
150(1)
7.5 Recurrence Relations
150(1)
7.6 Orthogonality
151(4)
7.7 Examples
155(14)
7.7.1 Quantum mechanical harmonic oscillator
155(4)
7.7.2 Optical fiber modes with a quadratic index variation
159(2)
7.7.3 Hermite-Gauss beams
161(5)
7.7.4 Relativistic Hermite polynomials
166(1)
7.7.5 Cortical receptive fields and vision
167(2)
7.8 Summary
169(2)
8 Gegenbauer, Jacobi, and Orthogonal Polynomials 171(14)
8.1 Introduction
171(1)
8.2 Gegenbauer Polynomials
171(4)
8.2.1 Relationship to other orthogonal polynomials
173(2)
8.3 Jacobi Polynomials
175(5)
8.3.1 Relationship to Gegenbauer polynomials
176(2)
8.3.2 Relationship to other polynomials
178(2)
8.4 Classical Orthogonal Polynomial Functions and Differential Equation
180(3)
8.4.1 Common properties
180(2)
8.4.2 Alternative forms of the general differential equation
182(1)
8.5 Summary
183(2)
9 Laguerre Polynomials 185(20)
9.1 Introduction
185(1)
9.2 Series Solution
185(3)
9.3 Generating Function and Recurrence Relations
188(2)
9.4 Orthogonality
190(2)
9.5 Integral Relationships
192(1)
9.6 Associated Laguerre Polynomials
193(4)
9.6.1 Generating function
194(2)
9.6.2 Rodrigues' formula and other relations
196(1)
9.7 A Warning
197(1)
9.8 Examples
197(6)
9.8.1 Optical fiber
198(2)
9.8.2 Laguerre-Gauss beams
200(3)
9.8.3 Data compression
203(1)
9.9 Summary
203(2)
10 Legendre Functions 205(38)
10.1 Introduction
205(1)
10.2 Series Solution
205(3)
10.3 Generating Function and Recurrence Relations
208(5)
10.3.1 Legendre polynomials in trigonometric form
211(2)
10.4 Rodrigues' Formula
213(1)
10.5 Integral Representation of Legendre Polynomials
214(3)
10.6 Orthogonality
217(2)
10.7 Associated Legendre Functions
219(2)
10.8 Other Properties of the Associated Legendre Function
221(1)
10.8.1 Orthogonality
221(1)
10.8.2 Recurrence relations
221(1)
10.8.3 Integral relationships
221(1)
10.9 Spherical Harmonics
221(12)
10.9.1 A note on (-1)m
224(1)
10.9.2 Some properties of Y1m(theta,phi)
225(2)
10.9.3 Spherical harmonics addition theorem
227(6)
10.10 Vector Spherical Harmonics
233(4)
10.11 Examples
237(4)
10.11.1 Multipole expansions
237(2)
10.11.2 Geomagnetics
239(2)
10.11.3 Computer graphics
241(1)
10.12 Summary
241(2)
11 Mathieu Functions 243(26)
11.1 Introduction
243(1)
11.2 Elliptical Coordinate System
243(4)
11.3 Mathieu Differential Equation(s)
247(1)
11.4 Angular Mathieu Function
248(1)
11.4.1 Floquet's theorem
248(1)
11.4.2 Hill equation
249(1)
11.5 Series Solutions to the Mathieu Equation
249(3)
11.6 Recurrence Relations and Other Factors
252(1)
11.7 Evaluation of an and bn
253(2)
11.8 Modified Mathieu Functions
255(1)
11.9 List of Relationships and Identities
256(2)
11.9.1 Relationship to Bessel functions
256(1)
11.9.2 Modified Mathieu function identities
257(1)
11.9.3 Asymptotic expansions of the radial Mathieu functions
257(1)
11.9.4 Some derivative relationships
257(1)
11.10 Nomenclature
258(1)
11.11 A Note on Python Coding
258(2)
11.12 Examples
260(6)
11.12.1 Quantum pendulum
260(2)
11.12.2 Mathieu beams
262(2)
11.12.3 Nanoantennas
264(1)
11.12.4 Paul traps
264(2)
11.13 Summary
266(3)
12 Hypergeometric Functions 269(22)
12.1 Introduction
269(1)
12.2 The Hypergeometric Function: Power Series Solution
270(1)
12.3 Pochhammer Symbol
271(1)
12.4 Indicial Equations for Hypergeometric Functions
272(4)
12.4.1 Case 1
272(1)
12.4.1.1 p= 0
272(1)
12.4.1.2 p = 1-c
273(1)
12.4.2 Case 2: c = 1
273(1)
12.4.3 Case 3: c = 0, ±1, ±2, ±3, ±4...
274(2)
12.5 Some Properties of Hypergeometric Functions
276(2)
12.6 Solutions to the Hypergeometric Equation
278(1)
12.7 The Confluent Hypergeometric Equation
279(2)
12.8 Some Properties of the Confluent Hypergeometric Function
281(1)
12.9 Relationship of 2F1 and 1F1 Functions to Other Functions
282(4)
12.9.1 Relationship to elementary functions
282(1)
12.9.2 Relationship to special functions
283(3)
12.10 Asymptotic Expansions
286(1)
12.11 Whittaker Functions
287(2)
12.12 Summary
289(2)
13 Integral Transforms 291(30)
13.1 Introduction
291(9)
13.1.1 Appearance of an integral transform
292(3)
13.1.2 General integral transforms
295(1)
13.1.3 Some properties of integral transforms
296(3)
13.1.4 Summary
299(1)
13.2 Hankel Transforms
300(5)
13.2.1 Relationship to Fourier transform
302(1)
13.2.2 Examples
303(2)
13.2.2.1 Laplace equation
303(1)
13.2.2.2 Laser propagation
304(1)
13.3 Fresnel Transforms
305(9)
13.3.1 Definitions and basic relationships
305(3)
13.3.2 Fresnel zone plates
308(6)
13.3.2.1 Zone plate focus
309(2)
13.3.2.2 Resolution of Fresnel zone plate
311(1)
13.3.2.3 Zone plate and source bandwidth
312(1)
13.3.2.4 Application of Fresnel zone plates
313(1)
13.3.3 Comparison between the Dirac delta function, the Fourier transform, and the Fresnel transform
314(1)
13.4 The Wigner Function
314(6)
13.4.1 Definition
315(1)
13.4.2 Properties of the Wigner function
316(1)
13.4.3 Some examples of Wigner distribution functions
317(2)
13.4.3.1 Delta function
317(1)
13.4.3.2 Harmonic function
317(1)
13.4.3.3 Spherical wave
318(1)
13.4.3.4 Gaussian signal
318(1)
13.4.3.5 Hermite-Gauss beams
318(1)
13.4.4 Two applications
319(1)
13.4.4.1 Radiometry
319(1)
13.4.4.2 Optical aberrations
319(1)
13.5 Summary
320(1)
14 Zernike Polynomials 321(18)
14.1 Introduction
321(1)
14.2 Description of Zernike Polynomials
321(4)
14.3 Indexing Schemes
325(1)
14.4 Python Codes for Zernike Polynomials
325(1)
14.5 Integral Representation and Orthonormality of Zernike Polynomials
326(1)
14.6 Recurrence Relations and Derivatives
327(1)
14.7 Relationship to Other Special Functions
328(3)
14.8 Relationship to Taylor Series and Seidel Aberrations
331(1)
14.9 Primary Aberrations
332(2)
14.10 Wavefront Error
334(1)
14.11 Some Advantages of Using Zernike Polynomials
334(3)
14.12 Conclusion
337(2)
Appendix A: Series Solution of Differential Equations 339(6)
Appendix B: Python Basics 345(16)
Appendix C: Additional Reading 361(4)
Postscript 365(2)
References 367(14)
Index 381