Uniform Stationary Phase Method

Cover......Page 1
Frontmatter......Page 2
Contents......Page 4
Preface......Page 6
Introduction: Asymptotic expansions......Page 10
1 Asymptotic contribution of critical point in one-dimensional integral......Page 16
1.1 Asymptotic contribution of integration domain boundary......Page 17
1.2 Asymptotic contribution of phase function stationary point......Page 18
1.3 Isolated critical points and partition of unity......Page 24
1.4 Branch point contribution......Page 26
1.5 One-sided and two-sided integrals......Page 29
1.6 Asymptotic contribution of pole of nonexponential factor......Page 32
1.7 Integrals of functions with large parameter in argument......Page 33
2 Two critical points merging in a one-dimensional integral......Page 38
2.1 Stationary point near integration domain boundary......Page 39
2.2 Stationary point near pole......Page 45
2.3 Two close stationary points......Page 50
2.4 Close stationary and branch points......Page 57
2.5 Close pole and branch points......Page 62
2.6 Pole near integration domain boundary......Page 64
2.7 Stationary point tending to infinity......Page 67
2.8 Applicability range for nonuniform asymptotics......Page 72
2.8.6 Stationary point tending to infinity in integral of eqn. 2.64 type......Page 73
3 Three close critical points in one-dimensional integral......Page 75
3.1 Three close stationary points......Page 76
3.2 Two stationary points near integration domain boundary......Page 79
3.3 Two stationary points near pole......Page 82
3.4 Pole and stationary point near integration domain boundary......Page 84
4.1.1 Partition of unity for two-fold integrals......Page 88
4.1.2 Asymptotic contribution of nondegenerate stationary point......Page 90
4.1.3 Critical point on smooth boundary segment......Page 92
4.1.4 Corner point of the integration domain boundary......Page 93
4.2.1 Close stationary points of phase function......Page 95
4.2.2 Critical point on boundary near corner point......Page 97
4.2.3 Two close critical points on smooth part of boundary......Page 99
4.2.4 Phase function stationary point near smooth part of boundary......Page 100
4.3 Stationary saddle point near boundary......Page 103
4.3.1 Corner point of boundary near stationary point......Page 105
4.3.2 Smooth boundary Σ close to saddle stationary point O......Page 108
4.3.3 Proof of eqn. 4.28......Page 111
4.4 Extremum point of phase function near corner point of boundary......Page 114
4.5 Proof of eqn. 4.42......Page 118
4.6 Influence domain for nondegenerate stationary point......Page 125
5.1 Cylindrical functions and their asymptotics......Page 127
5.2 Reflection from smooth surface in Kirchhoff approximation......Page 132
5.3 Diffraction by wedge: asymptotic of exact solution......Page 137
5.4 Diffraction by wedge and slit: physical optics approximation......Page 141
5.5 Cylindrical wave reflection from plane interface of two media......Page 144
5.6 Reflection from concave surface......Page 152
5.7 Radiation from aperture (nonuniform asymptotic)......Page 157
5.8 Radiation from aperture (uniform asymptotic)......Page 162
5.9 Diffraction by two wedges......Page 164
6.1 Fresnel integral......Page 168
6.2 Airy functions......Page 170
6.3 Parabolic cylinder functions......Page 172
6.4 Incomplete gamma function......Page 174
6.5 Integral sine and cosine......Page 175
6.6 Hankel functions......Page 176
6.7 Piercey integral......Page 177
6.8 Incomplete Airy function......Page 179
6.9 Airy-Fresnel integral......Page 182
6.10 Generalised Fresnel integral......Page 185
6.11 Ff-integral......Page 186
References......Page 191
Appendix: Coefficients of special function expansions in Chebyshev's polynomials......Page 193
A.1 Fresnel integrals......Page 196
A.2 Parabolic cylinder functions......Page 198
A.3 Integral sine and cosine......Page 206
A.4 Bessel functions......Page 208
A.5 Bessel functions of imaginary argument......Page 220
A.6 Airy-Fresnel integral......Page 224
A.7 Generalised Fresnel integral......Page 229
A.8 Ff-integral......Page 233
Index......Page 241