Abstract

Wolf’s result for integrated flux in the case of diffraction by a circular lens or aperture in the scalar, paraxial Fresnel approximation is considered anew. Compact integral formulas for pertinent infinite sums are derived, and the result’s generalizations to extended sources and Planckian sources and asymptotic aspects at small wavelength and high temperature are all considered. Simplification of calculations for an actual absolute radiometer is demonstrated.

© 2004 Optical Society of America

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References

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  1. E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).
  2. E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
    [CrossRef]
  3. J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
    [CrossRef]
  4. W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
    [CrossRef]
  5. W. H. Steel, M. De, J. A. Bell, “Diffraction corrections in radiometry,” J. Opt. Soc. Am. 62, 1099–1103 (1972).
    [CrossRef]
  6. L. P. Boivin, “Diffraction corrections in radiometry: comparison of two different methods of calculation,” Appl. Opt. 14, 2002–2009 (1975).
    [CrossRef] [PubMed]
  7. E. L. Shirley, “Revised formulas for diffraction effects with point and extended sources,” Appl. Opt. 37, 6581–6590 (1998).
    [CrossRef]
  8. P. Edwards, M. McCall, “Diffraction loss in radiometry,” Appl. Opt. 42, 5024–5032 (2003).
    [CrossRef] [PubMed]
  9. E. L. Shirley, “Diffraction effects on broadband radiation: formulation for computing total irradiance,” Appl. Opt. 43, 2609–2620 (2004).
    [CrossRef] [PubMed]
  10. F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), pp. 237–238.
  11. U. J. Knottnerus, Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters (Wolters, Groningen, The Netherlands, 1960), pp. 34–35.
  12. Ref. 10, p. 293.
  13. N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996), p. 59.
  14. Ref. 13, p. 46.
  15. R. W. Brusa, C. Fröhlich, “Absolute radiometers (PMO6) and their experimental characterization,” Appl. Opt. 25, 4173–4180 (1986).
    [CrossRef]

2004 (1)

2003 (1)

1998 (1)

1986 (1)

1975 (1)

1972 (1)

1970 (1)

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

1956 (1)

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

1951 (1)

E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
[CrossRef]

1885 (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Bell, J. A.

Blevin, W. R.

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

Boivin, L. P.

Brusa, R. W.

De, M.

Edwards, P.

Focke, J.

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

Fröhlich, C.

Knottnerus, U. J.

U. J. Knottnerus, Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters (Wolters, Groningen, The Netherlands, 1960), pp. 34–35.

Lommel, E.

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

McCall, M.

Olver, F. W. J.

F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), pp. 237–238.

Shirley, E. L.

Steel, W. H.

Temme, N. M.

N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996), p. 59.

Wolf, E.

E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
[CrossRef]

Abh. Bayer. Akad. (1)

E. Lommel, “Die Beugungserscheinungen einer kreisrunden Oeffnung und eines kreisrunden Schirmschens theoretisch und experimentell Bearbeitet,” Abh. Bayer. Akad. 15, 233–328 (1885).

Appl. Opt. (5)

J. Opt. Soc. Am. (1)

Metrologia (1)

W. R. Blevin, “Diffraction losses in radiometry and photometry,” Metrologia 6, 39–44 (1970).
[CrossRef]

Opt. Acta (1)

J. Focke, “Total illumination in an aberration-free diffraction image,” Opt. Acta 3, 161–163 (1956).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

E. Wolf, “Light distribution near focus in an error-free diffraction image,” Proc. R. Soc. London Ser. A 204, 533–548 (1951).
[CrossRef]

Other (5)

F. W. J. Olver, Asymptotics and Special Functions (Peters, Wellesley, Mass., 1997), pp. 237–238.

U. J. Knottnerus, Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters (Wolters, Groningen, The Netherlands, 1960), pp. 34–35.

Ref. 10, p. 293.

N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics (Wiley, New York, 1996), p. 59.

Ref. 13, p. 46.

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Figures (2)

Fig. 1
Fig. 1

Class of optical setup considered in this work. The aperture can be limiting (as shown in this case) or nonlimiting.

Fig. 2
Fig. 2

Contour integration for Barnes’s integral for hypergeometric function discussed in text, for s=0.

Tables (3)

Tables Icon

Table 1 Values and Derivatives of Riemann Zeta Function for Lowest Even Positive Integers

Tables Icon

Table 2 Nontrivial Values of Cs,p for Small s and p

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Table 3 Nontrivial Values of Ls,p for Small s and p

Equations (110)

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u=(2πRa2/λ)(1/ds+1/dd-1/f ),
vs=(2π/λ)(RsRa/ds),
vd=(2π/λ)(RdRa/dd),
v0=max(vs, vd),
σ=min(vs, vd)/max(vs,vd).
Q2s(v)=p=02s(-1)p[Jp(v)J2s-p(v)+Jp+1(v)J2s+1-p(v)]
Yn(u, v)=s=0(-1)s(n+2s)(v/u)n+2sJn+2s(v),
LB(v, w)=s=0(-1)sw2sQ2s(v)/(2s+1)
LX(v, w)=(4w/v)[Y1(v/w, v)cos(gv)+Y2(v/w, v)sin(gv)],
Φλ(λ)=C-11dx(1+σx)-1{(1-x2)[(2+σx)2-σ2]}1/2L(u, v0(1+σx))Lλ(λ),
L(u, v0(1+σx))=L(u, v)=1-LB(v, w)=1-LB(α/λ, w)
L(u, v0(1+σx))=L(u, v)=w2[1+LB(v, w)]-LX(v, w)=w2[1+LB(α/λ, w)]-LX(a/λ, w),
Φ=C-11dx(1+σx)-1{(1-x2)[(2+σx)2-σ2]}1/20dλL(u, v)Lλ(λ),
Lλ(λ)=c1πλ5expc2λT-1-1.
A=c2/(αT)
0dλL(u, v)Lλ(λ)=c1πα40dvv3exp(Av)-1×[1-LB(v, w)]
=c1πα46ζ(4)A4-FB(A, w)
0dλL(u, v)Lλ(λ)
=c1πα40dvv3exp(Av)-1 {w2[1+LB(v, w)]-LX(v, w)}=c1πα46w2ζ(4)A4+w2FB(A, w)-FX(A, w).
FB(A, w)=0dvv3exp(Av)-1 LB(v, w),
FX(A, w)=0dvv3exp(Av)-1 LX(v, w),
Jm(v)=(-i)m2π02πdθ exp(ivx+imθ),
p=02s(-1)pJp(v)J2s-p(v)
=(-1)sp=-ss(-1)pJs+p(v)Js-p(v)=(-1)sp=-ss(-1)pi-s-p2π02πdθ exp(ivx)×exp[i(s+p)θ]
×i-s+p2π02πdθexp(ivx)exp[i(s-p)θ].
p=02s(-1)pJp(v)J2s-p(v)
=(-1)sp=-ss(-1)pi-s-p2π02πdθ×exp[i(vx+sθ+pθ)]×is-p2π02πdθexp[i(-vx-sθ+pθ)]=(-1)s(2π)2p=-ss02πdθ exp[i(vx+sθ+pθ)]×02πdθexp[i(-vx-sθ+pθ)]=(-1)s(2π)202πdθ exp(ivx)02πdθexp(-ivx)×ηsh2s+1-h-2s-1h-h-1
p=02s(-1)pJp+1(v)J2s+1-p(v)
=(-1)s(2π)202πdθ exp(ivx)02πdθexp(-ivx)×ηs+1h2s+1-h-2s-1h-h-1,
Q2s(v)=(-1)s(2π)202πdθ exp(ivx)02πdθexp(-ivx)×1+ηh-h-1[h(ηh2)s-h-1(ηh-2)s].
LB(v, w)=s=0(-1)sw2sQ2s(v)/(2s+1)=1(2π)202πdθ exp(ivx)02πdθexp(-ivx)×1+ηh-h-1hs=0w12s2s+1-h-1s=0w22s2s+1=1(2π)202πdθ exp(ivx)02πdθexp(-ivx)×1+ηh-h-1h2w1loge1+w11-w1-h-12w2loge1+w21-w2.
FB(A, w)=0dvv3exp(Av)-1 LB(v, w)=120dvv3exp(Av)-1 [LB(v, w)+LB(-v, w)]=3(2π)202πdθ02πdθ1+ηh-h-1×h2w1loge1+w11-w1-h-12w2loge1+w21-w2S(A, x-x),
S(x, y)=S(x, -y)=n=1[(nx+iy)-4+(nx-iy)-4].
S(x, y)=-(2π/x)4[1/z4+(f-7f2+12 f3-6f4)/6].
S(x, y)=32π4/x4[1/1440-z2/6048+z4/69120-].
Jm(v)(2/πv)1/2cos ζs=0(-1)sA2s(m)v-2s-sin ζs=0(-1)sA2s+1(m)v-(2s+1),
Q2s(v)=(-1)s(2s+1)2πv-cos 2vπv2-16s4+32s3+8s2-8s-312πv3+8s2+8s-14πv3sin(2v)+64s4+128s3-16s2-80s+932πv4cos(2v)+O(v-5).
σk=s=0skw2s=w2dd(w2)k11-w2=Wk(w2)(1-w2)k+1,
W0(x)=1,W3(x)=x3+4x2+x,
W1(x)=x,W4(x)=x4+11x3+11x2+x,
W2(x)=x2+x,
W5(x)=x5+26x4+66x3+26x2+x.
LB(v, w)=s=0(-1)sw2sQ2s(v)/(2s+1)=2σ0πv-σ0cos(2v)πv2-16σ4+32σ3+8σ2-8σ1-3σ012πv3+8σ2+8σ1-σ04πv3sin(2v)+64σ4+128σ3-16σ2-80σ1+9σ032πv4×cos(2v)+O(v-5).
I2s(A)=0dvv3exp(Av)-1 Q2s(v)=n=0i2s(nA),
i2s(A)=0dvv3exp(-Av)Q2s(v),
Ja(v)Jb(v)=v2a+br=0s=0(-v2/4)r+sr!s!(r+a)!(s+b)!=v2a+bm=0-v24mk=0m[k!(m-k)!(k+a)!(m+b-k)!]-1=v2a+bm=0-v24mk=0m{k!(m+b-k)!(m-k)![m+a-(m-k)]!}-1=v2a+bm=0-v24m[(m+a)!(m+b)!]-1k=0mm+bkm+am-k=v2a+bm=0-v24m[(m+a)!(m+b)!]-12m+a+bm.
0dvv3+a+b+2mexp(-Av)=Γ(2m+a+b+4)A2m+a+b+4
0dvv3exp(-Av)Ja(v)Jb(v)
=12a+bAa+b+4m=0-14A2m×Γ(2m+a+b+4)Γ(2m+a+b+1)(m+a)!(m+b)!m!Γ(m+a+b+1).
 i2s(A)=c=01122s+2cA2s+2c+4m=0-14A2mΓ(2m+2s+2c+4)Γ(2m+2s+2c+1)T(m+c, 2s)m!Γ(m+2s+2c+1),
T(M, 2s)=p=02s(-1)p[(M+p)!(M+2s-p)!]-1=(-1)sp=-ss(-1)p[(M+s+p)!(M+s-p)!]-1.
T(M, 2s)|M>0=[(M+s)Γ(M)Γ(1+M+2s)]-1.
T(0, 2s)=1(2s)!p=02s2sp(-1)p=(1-1)2s(2s)!=δs,0,
T(M, 2s)|M>0=(-1)M(2M+2s)!p=-ss02πdθ2πexp(2ipθ)×[exp(iθ)-exp(-iθ)]2M+2s.
 i2s(A)=6A-4δs,0+1A4(2A)2sm=0z(-z)m×Γ(2m+2s+6)Γ(2m+2s+3)m!Γ(m+2s+3)(m+1+s)Γ(m+1)Γ(m+2s+2)+1A4(2s)2sm=0(-z)m+1Γ(2m+2s+6)Γ(2m+2s+3)(m+1)!Γ(m+2s+2)(m+1+s)Γ(m+1)Γ(m+2s+2),
i2s(A)=6A-4δs,0+zA4(2A)2sm=0(-z)mm!Γ(2m+2s+6)Γ(2m+2s+3)(m+1+s)Γ(m+2s+2)1m!Γ(m+2s+3)-1(m+1)!Γ(m+2s+2)=6A-4δs,0+222s+2A2s+6m=0(-z)mΓ(2m+2s+6)Γ(2m+2s+2)m!Γ(m+2s+2)(m+1)-(m+2s+2)(m+1)!Γ(m+2s+3)=6A-4δs,0-2(2s+1)22s+2A2s+6m=0(-z)mm!Γ(2m+2s+6)Γ(2m+2s+2)Γ(m+2)Γ(m+2s+2)Γ(m+2s+3).
i2s(A)=6A-4δs,0-2(2s+1)Γ(2s+6)Γ(2s+2)22s+2A2s+6m=0(-z)mm!24m(s+3)m(s+7/2)m(s+1)m(s+3/2)mΓ(2)Γ(2s+2)Γ(2s+3)(2)m(2s+2)m(2s+3)m.
i2s(A)=6A-4δs,0-2(2s+1)(2s+3)(2s+4)(2s+5)22s+2A2s+6m=0(-4/A2)mm!(s+3)m(s+7/2)m(s+1)m(s+3/2)m(2)m(2s+2)m(2s+3)m=6A-4δs,0-2(2s+1)(2s+3)(2s+4)(2s+5)22s+2A2s+6×4F3(s+1, s+3/2, s+3, s+7/2;2, 2s+2, 2s+3;-4/A2).
4F3(s+1, s+3/2, s+3, s+7/2;2, 2s+2, 2s+3;-4/A2)=12πiΓ(2)Γ(2s+2)Γ(2s+3)Γ(s+1)Γ(s+3/2)Γ(s+3)Γ(s+7/2)
×Ddt Γ(s+1+t)Γ(s+3/2+t)Γ(s+3+t)Γ(s+7/2+t)Γ(-t)exp(λt)Γ(2+t)Γ(2s+2+t)Γ(2s+3+t).
2(2s+1)(2s+3)(2s+4)(2s+5)Γ(2s+2)Γ(2s+3)22s+2A2s+6Γ(s+1)Γ(s+3/2)Γ(s+3)Γ(s+7/2)=2(2s+1)22s+2A2s+6Γ(2s+2)Γ(s+1)Γ(s+3/2)Γ(2s+6)Γ(s+3)Γ(s+7/2)=2(2s+1)22s+2A2s+622(s+1)-1Γ(1/2)22(s+3)-1Γ(1/2)=2s+12π2A2s+6,
i2s(A)=6A4 δs,0-2s+12π2A2s+612πiDdt Γ(s+1+t)Γ(s+3/2+t)Γ(s+3+t)Γ(s+7/2+t)Γ(-t)exp(λt)Γ(2+t)Γ(2s+2+t)Γ(2s+3+t).
I2s(A)=6ζ(4)δs,0A4-(2s+1)4s+34π2iA2s+6Ddt Γ(s+1+t)Γ(s+3/2+t)Γ(s+3+t)Γ(s+7/2+t)Γ(-t)ζ(2s+2t+6)exp(λt)Γ(2+t)Γ(2s+2+t)Γ(2s+3+t).
I2s(A)p=-4(Cs,p+Ls,ploge A)Ap.
Γ(z+)=Γ(z)[1+ψ(z)]+O(2),
Γ(z+)=Γ(-N+)=[(-1)N/N!][1/+ψ(N+1)]+O().
ζ(z+)=ζ(z)+ζ(z)+O(2),
ζ(z+)=ζ(1+)=1/+γ+O().
exp[λ(z+)]=[1+λ+O(2)]exp(λz)
-12π2A6Γ(1/2)Γ(2)Γ(5/2)Γ(1)ζ(4)(4/A2)-1Γ(1)Γ(1)Γ(2)
=-6ζ(4)A4.
Cs,0=(-1)s+1(2s+1)(s6+3s5+s4-3s3-2s2)/6.
s=0(-1)sw2sCs,-3A-3/(2s+1)
=4ζ(3)A-3/[π(1-w2)],
Cs,-1=-(2s+3)(2s+1)3(2s-1)(-1)s24π×-2γ-2 loge 2+113-ψs+52-ψs+32-ψs+12-ψs-12
Ls,-1=+(2s+3)(2s+1)3(2s-1)(-1)s12π.
Pτ=(-1)τ+1ζ(-1-2τ)Γ(-5/2-τ)Γ(-1/2-τ)(2π)222ττ!(2+τ)!=8ζ(2+2τ)π3+2τΓ(6+2τ),
Qs,τ=(-1)s(2s+1)Γ(s+7/2+τ)Γ(s+5/2+τ)Γ(s-1/2-τ)Γ(s-3/2-τ)=(-1)s(2s+1)28+4τ (2s+2τ+5)[(2s+2τ+3)×(2s-2τ-1)]2(2s-2τ-3),
Cs,1+2τ=PτQs,τ[ψ(-5/2-τ)+ψ(-1/2-τ)+ψ(3+τ)+ψ(1+τ)+2 loge 2+2ζ(-1-2τ)/ζ(-1-2τ)-ψ(s+7/2+τ)-ψ(s+5/2+τ)-ψ(s-1/2-τ)-ψ(s-3/2-τ)]
ζ(1-z)/ζ(1-z)=loge 2π-ψ(z)+(π/2)tan(πz/2)-ζ(z)/ζ(z),
2[Y1(v/w, v)cos(gv)+Y2(v/w, v)sin(gv)]
=exp(-igv)s=0(-1)sw1+2s(1+2s)J1+2s(v)+is=0(-1)sw2+2s(2+2s)J2+2s(v)+c.c.
2[Y1(v/w, v)cos(gv)+Y2(v/w, v)sin(g/v)]=v exp(-igv)2s=0(-1)sw1+2s[J2s(v)+J2+2s(v)]+is=0(-1)sw2+2s[J1+2s(v)+J3+2s(v)]+c.c.=vw exp(-igv)2s=0(-w2)sJ2s(v)+iws=0(-w2)sJ1+2s(v)+s=0(-w2)sJ2+2s(v)+iws=0(-w2)sJ3+2s(v)+c.c.=vw exp(-igv)2s=0(iw)sJs(v)-s=0is+2wsJs+2(v)+c.c.=vw exp(-igv)4π02πdθ1-exp(2iθ)1-w exp(iθ)exp(ivx)+c.c.
2[Y1(v/w, v)cos(gv)+Y2(v/w, v)sin(gv)]
=vw exp(-igv)8π02πdθ1-exp(2iθ)1-w exp(iθ)+1-exp(-2iθ)1-w exp(-iθ)exp(ivx)+c.c.=vw4π02πdθ1-exp(2iθ)1-w exp(iθ)+1-exp(-2iθ)1-w exp(-iθ)cos[v(x-g)].
2[Y1(v/w, v)cos(gv)+Y2(v/w, v)sin(gv)]
=vwπ02πdθ1-x21+w2-2wxcos[v(x-g)].
LX(v, w)=2w2π02πdθ (1-x2)cos[v(x-g)]1+w2-2wx=4w2π0πdθ (1-x2)cos[v(x-g)]1+w2-2wx.
fx(A, w)
=w2π0dvv3exp(-Av)02πdθ1-x21+w2-2wx×exp[iv(x-g)]+c.c.=w4π-ddA3Cdziz4-(z+1/z)2(w+1/w)-(z+1/z)×1L-i(z+1/z)/2+c.c.=w2π-ddA3Cdzz×(z2-1)2(z-w)(z-1/w)(z2+1+2iLz)+c.c.
FX(A, w)=6w2π02πdθ (1-x2)S(A, g-x)1+w2-2wx=12w2π0πdθ (1-x2)S(A, g-x)1+w2-2wx.
Y1(v/w, v)=vw22πv1/22σ0+4σ1vsin(v-π/4)+3σ0+22σ1+48σ2+32σ34v2cos(v-π/4)+15σ0+62σ1-160σ2-960σ3-1280σ4-512σ564v3sin(v-π/4)+O(v-7/2),
Y2(v/w, v)=-v22πv1/24σ1vsin(v+π/4)+-σ1+16σ32v2cos(v+π/4)+-9σ1+160σ3-256σ532v3sin(v+π/4)+O(v-7/2).
z2+1+2iLz=[z+iL(1+R)][z+iL(1-R)]=z2+1+2z[iA-(w+1/w)/2]=[(z-w)(z-1/w)+2iAz].
(z2-1)2(z-w)(z-1/w)[(z-w)(z-1/w)+2iAz)z=0=1,
(z2-1)2z(z-1/w)[(z-w)(z-1/w)+2iAz)z=w
=(w2-1)2w(w-1/w)(2iAw)=w-1/w2iA.
(z2-1)2z(z-w)(z-1/w)[z+iL(1+R)]z=-iL(1-R)
=-4L2R2z2z(-2iAz)(2iLR)z=-iL(1-R)=-LR/A=-(L2+1)1/2/A.
fx(A, w)=iw-ddA31+w-1/w2iA-(L2+1)1/2A+c.c.=-ddA3w(w-1/w)A-iwA {[(A+ig)2+1]1/2-[(A-ig)2+1]1/2}.
[(A+ig)2+1]1/2=[(A+ig+i)(A+ig-i)]1/2=(A+ig+i)1/2(A+ig-i)1/2
(A+ig±i)1/2=(A+ig)1/2k=0(-1)kΓ(k-1/2)k!Γ(-1/2)×±iA+igk,
[(A+ig)2+1]1/2=(A+ig)k=0Ck(A+ig)-k=A+ig+k=0Ck+2k!dd(ig)k×1A+ig.
[(A-ig)2+1]1/2=(A-ig)k=0Ck(A-ig)-k=A-ig+k=0Ck+2k!dd(ig)k×1A-ig,
iwA {[(A+ig)2+1]1/2-[(A-ig)2+1]1/2}=-2gwA+iwk=0Ck+2k!dd(ig)k1A(A+ig)-1A(A-ig)=-2gwA+iwk=0Ck+2k!dd(ig)k1ig×2A-1A+ig-1A-ig,
fx(A, w)=6w(w-1/w)A4+2wA4×g-ik=0Ck+2k!dd(ig)k1ig+iwk=0Ck+2k!dd(ig)k1ig1(A+ig)4+1(A-ig)4.
2wA4g-ik=0Ck+2k!dd(ig)k1ig=-2iwA4ig+k=0Ck+2k!dd(ig)k1ig=-2iw[(ig)2+1]1/2/A4=2w(g2-1)1/2/A4.
fX(A, w)=6iwk=0Ck+2k!dd(ig)k1ig1(A+ig)4+1(A-ig)4,
FX(A, w)=6iwk=0Ck+2k!dd(ig)kS(A, g)ig.
FX(A, w)-iw4ddg4k=0Ck+2k!dd(ig)k1ig-iw4ddg4{[(ig)2+1]1/2-ig}±w4ddg4[g2-1]1/2-96w6(1+3w2+w4)(1-w2)7.

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