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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.28 Integrals, Integral Representations, and Integral Equations

Contents

§28.28(i) Equations with Elementary Kernels

Let

28.28.1 w=coshzcostcosα+sinhzsintsinα.

Then

28.28.2 12π02π2hwcen(t,h2)t=ncen(α,h2)Mcn(1)(z,h),
28.28.3 12π02π2hwsen(t,h2)t=nsen(α,h2)Msn(1)(z,h),
28.28.4 hπ02πwα2hwcen(t,h2)t=ncen(α,h2)Mcn(1)(z,h),
28.28.5 hπ02πwα2hwsen(t,h2)t=nsen(α,h2)Msn(1)(z,h).

In (28.28.7)–(28.28.9) the paths of integration j are given by

28.28.6 1 : from -η1+ to 2π-η1+,
3 : from -η1+ to η2-,
4 : from η2- to 2π-η1+,

where η1 and η2 are real constants.

28.28.7 1πj2hwmeν(t,h2)t=νπ/2meν(α,h2)Mν(j)(z,h),
j=3,4,
28.28.8 1πj2hwα2hwmeν(t,h2)t=νπ/2meν(α,h2)Mν(j)(z,h),
j=3,4,
28.28.9 12π12hwmeν(t,h2)t=νπ/2meν(α,h2)Mν(1)(z,h).

In (28.28.11)–(28.28.14)

28.28.10 0<ph(h(coshz±1))<π.
28.28.11 02hcoshzcoshtCeν(t,h2)t=12πνπceν(0,h2)Mν(3)(z,h),
28.28.12 02hcoshzcoshtsinhzsinhtSeν(t,h2)t=-π4hνπ/2seν(0,h2)Mν(3)(z,h),
28.28.13 02hcoshzcoshtsinhzsinhtFem(t,h2)t=-π4hmfem(0,h2)Mcm(3)(z,h),
28.28.14 02hcoshzcoshtGem(t,h2)t=12πm+1gem(0,h2)Msm(3)(z,h).

In particular, when h>0 the integrals (28.28.11), (28.28.14) converge absolutely and uniformly in the half strip z0, 0zπ.

28.28.15 0cos(2hcosycosht)Ce2n(t,h2)t=(-1)n+112πMc2n(2)(0,h)ce2n(y,h2),
28.28.16 0sin(2hcosycosht)Ce2n(t,h2)t=-πA02n(h2)2ce2n(12π,h2)(ce2n(y,h2)2πC2n(h2)fe2n(y,h2)),

where the upper or lower sign is taken according as 0yπ or πy2π. For A02n(q) and C2n(q) see §§28.4 and 28.5(i).

For details and further equations see Meixner et al. (1980, §2.1.1) and Sips (1970).

§28.28(ii) Integrals of Products with Bessel Functions

With the notations of §28.4 for Amn(q) and Bmn(q), §28.14 for cnν(q), and (28.23.1) for 𝒞μ(j), j=1,2,3,4,

28.28.17 1π0π𝒞ν+2s(j)(2hR)-(ν+2s)ϕmeν(t,h2)t=(-1)sc2sν(h2)Mν(j)(z,h),
s,

where R=R(z,t) and ϕ=ϕ(z,t) are analytic functions for z>0 and real t with

28.28.18 R(z,t) =(12(cosh(2z)+cos(2t)))1/2,
R(z,0) =coshz,

and

28.28.19 2ϕ =cosh(z+t)cosh(z-t),
ϕ(z,0) =0.

In particular, for integer ν and =0,1,2,,

28.28.20 2π0π𝒞2(j)(2hR)cos(2ϕ)ce2m(t,h2)t=ε(-1)+mA22m(h2)Mc2m(j)(z,h),

where again ε0=2 and ε=1, =1,2,3,.

28.28.21 2π0π𝒞2+1(j)(2hR)cos((2+1)ϕ)ce2m+1(t,h2)t=(-1)+mA2+12m+1(h2)Mc2m+1(j)(z,h),
28.28.22 2π0π𝒞2+1(j)(2hR)sin((2+1)ϕ)se2m+1(t,h2)t=(-1)+mB2+12m+1(h2)Ms2m+1(j)(z,h),
28.28.23 2π0π𝒞2+2(j)(2hR)sin((2+2)ϕ)se2m+2(t,h2)t=(-1)+mB2+22m+2(h2)Ms2m+2(j)(z,h).

§28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order

With the parameter h suppressed we use the notation

28.28.24 D0(ν,μ,z) =Mν(3)(z)Mμ(4)(z)-Mν(4)(z)Mμ(3)(z),
D1(ν,μ,z) =Mν(3)(z)Mμ(4)(z)-Mν(4)(z)Mμ(3)(z),

and assume ν and m. Then

28.28.25 sinhzπ202πcostmeν(t,h2)me-ν-2m-1(t,h2)sinh2z+sin2tt=(-1)m+1hαν,m(0)D0(ν,ν+2m+1,z),
28.28.26 coshzπ202πsintmeν(t,h2)me-ν-2m-1(t,h2)sinh2z+sin2tt=(-1)m+1hαν,m(1)D0(ν,ν+2m+1,z),

where

28.28.27 αν,m(0)=12π02πcostmeν(t,h2)me-ν-2m-1(t,h2)t=(-1)m2πmeν(0,h2)me-ν-2m-1(0,h2)hD0(ν,ν+2m+1,0),
28.28.28 αν,m(1)=12π02πsintmeν(t,h2)me-ν-2m-1(t,h2)t=(-1)m+12πmeν(0,h2)me-ν-2m-1(0,h2)hD1(ν,ν+2m+1,0).
28.28.29 coshzπ202πsintmeν(t,h2)me-ν-2m-1(t,h2)sinh2z+sin2tt=(-1)m+1hαν,m(0)D1(ν,ν+2m+1,z),
28.28.30 sinhzπ202πcostmeν(t,h2)me-ν-2m-1(t,h2)sinh2z+sin2tt=(-1)mhαν,m(1)D1(ν,ν+2m+1,z),
28.28.31 2π202πcostsintmeν(t,h2)me-ν-2m(t,h2)sinh2z+sin2tt=(-1)mγν,mD0(ν,ν+2m,z),
28.28.32 sinh(2z)π202πmeν(t,h2)me-ν-2m(t,h2)sinh2z+sin2tt=(-1)m+1γν,mD1(ν,ν+2m,z),

where

28.28.33 γν,m=12π02πmeν(t)me-ν-2m(t)t=(-1)m4πmeν(0)me-ν-2m(0)D1(ν,ν+2m,0).

Also,

28.28.34 1π202πmeν(t,h2)me-ν-2m-1(t,h2)sintt=(-1)m+1hαν,m(0)D1(ν,ν+2m+1,0),

where the integral is a Cauchy principal value (§1.4(v)).

§28.28(iv) Integrals of Products of Mathieu Functions of Integer Order

Again with the parameter h suppressed, let

28.28.35 Ds0(n,m,z) =Msn(3)(z)Msm(4)(z)-Msn(4)(z)Msm(3)(z),
Ds1(n,m,z) =Msn(3)(z)Msm(4)(z)-Msn(4)(z)Msm(3)(z),
Ds2(n,m,z) =Msn(3)(z)Msm(4)(z)-Msn(4)(z)Msm(3)(z).

Then

28.28.36 sinhzπ202πcostsen(t,h2)sem(t,h2)sinh2z+sin2tt=(-1)p+1hα^n,m(s)Ds0(n,m,z),
28.28.37 coshzπ202πsintsen(t,h2)sem(t,h2)sinh2z+sin2tt=(-1)p+1hα^n,m(s)Ds1(n,m,z),

where m-n=2p+1, p; m,n=1,2,3,. Also,

28.28.38 α^n,m(s)=12π02πcostsen(t,h2)sem(t,h2)t=(-1)p2πsen(0,h2)sem(0,h2)hDs2(n,m,0).

Let

28.28.39 Dc0(n,m,z) =Mcn(3)(z)Mcm(4)(z)-Mcn(4)(z)Mcm(3)(z),
Dc1(n,m,z) =Mcn(3)(z)Mcm(4)(z)-Mcn(4)(z)Mcm(3)(z),
28.28.40 Dsc0(n,m,z) =Msn(3)(z)Mcm(4)(z)-Msn(4)(z)Mcm(3)(z),
Dsc1(n,m,z) =Msn(3)(z)Mcm(4)(z)-Msn(4)(z)Mcm(3)(z).

Then

28.28.41 coshzπ202πsintsen(t,h2)cem(t,h2)sinh2z+sin2tt=(-1)p+1hβ^n,mDsc0(n,m,z),
28.28.42 sinhzπ202πcostsen(t,h2)cem(t,h2)sinh2z+sin2tt=(-1)phβ^n,mDsc1(n,m,z),

where m-n=2p+1, p; m=0,1,2,, n=1,2,3,. Also,

28.28.43 β^n,m=12π02πsintsen(t,h2)cem(t,h2)t=(-1)p2πsen(0,h2)cem(0,h2)hDsc1(n,m,0).

Next,

28.28.44 1π202πsin(2t)sen(t,h2)cem(t,h2)sinh2z+sin2tt=(-1)pγ^n,mDsc0(n,m,z),
28.28.45 sinh(2z)π202πsen(t,h2)cem(t,h2)sinh2z+sin2tt=(-1)p+1γ^n,mDsc1(n,m,z),

where n-m=2p, p; m=0,1,2,, n=1,2,3,. Also,

28.28.46 γ^n,m=12π02πsen(t,h2)cem(t,h2)t=(-1)p+14πsen(0,h2)cem(0,h2)Dsc1(n,m,0).

Lastly,

28.28.47 sinhzπ202πcostcen(t,h2)cem(t,h2)sinh2z+sin2tt=(-1)p+1hα^n,m(c)Dc0(n,m,z),
28.28.48 coshzπ202πsintcen(t,h2)cem(t,h2)sinh2z+sin2tt=(-1)p+1hα^n,m(c)Dc1(n,m,z),

where m-n=2p+1, p; m,n=0,1,2,. Also,

28.28.49 α^n,m(c)=12π02πcostcen(t,h2)cem(t,h2)t=(-1)p+12πcen(0,h2)cem(0,h2)hDc0(n,m,0).

§28.28(v) Compendia

See Prudnikov et al. (1990, pp. 359–368), Gradshteyn and Ryzhik (2000, pp. 755–759), Sips (1970), and Meixner et al. (1980, §2.1.1).