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28 Mathieu Functions and Hill’s EquationModified Mathieu Functions

§28.28 Integrals, Integral Representations, and Integral Equations

Contents
  1. §28.28(i) Equations with Elementary Kernels
  2. §28.28(ii) Integrals of Products with Bessel Functions
  3. §28.28(iii) Integrals of Products of Mathieu Functions of Noninteger Order
  4. §28.28(iv) Integrals of Products of Mathieu Functions of Integer Order
  5. §28.28(v) Compendia

§28.28(i) Equations with Elementary Kernels

Let

28.28.1 w=coshzcostcosα+sinhzsintsinα.

Then

28.28.2 12π02πe2ihwcen(t,h2)dt=incen(α,h2)Mcn(1)(z,h),
28.28.3 12π02πe2ihwsen(t,h2)dt=insen(α,h2)Msn(1)(z,h),
28.28.4 ihπ02πwαe2ihwcen(t,h2)dt=incen(α,h2)Mcn(1)(z,h),
28.28.5 ihπ02πwαe2ihwsen(t,h2)dt=insen(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+i to 2πη1+i,
3 : from η1+i to η2i,
4 : from η2i to 2πη1+i,

where η1 and η2 are real constants.

28.28.7 1πje2ihwmeν(t,h2)dt=eiνπ/2meν(α,h2)Mν(j)(z,h),
j=3,4,
28.28.8 1πj2ihwαe2ihwmeν(t,h2)dt=eiνπ/2meν(α,h2)Mν(j)(z,h),
j=3,4,
28.28.9 12π1e2ihwmeν(t,h2)dt=eiνπ/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 0e2ihcoshzcoshtCeν(t,h2)dt=12πieiνπceν(0,h2)Mν(3)(z,h),
28.28.12 0e2ihcoshzcoshtsinhzsinhtSeν(t,h2)dt=π4heiνπ/2seν(0,h2)Mν(3)(z,h),
28.28.13 0e2ihcoshzcoshtsinhzsinhtFem(t,h2)dt=π4himfem(0,h2)Mcm(3)(z,h),
28.28.14 0e2ihcoshzcoshtGem(t,h2)dt=12πim+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.

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)ei(ν+2s)ϕmeν(t,h2)dt=(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 e2iϕ =cosh(z+it)cosh(zit),
ϕ(z,0) =0.

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

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

where again ε0=2 and ε=1, =1,2,3,.

28.28.21 4π0π/2𝒞2+1(j)(2hR)cos((2+1)ϕ)ce2m+1(t,h2)dt=(1)+mA2+12m+1(h2)Mc2m+1(j)(z,h),
28.28.22 4π0π/2𝒞2+1(j)(2hR)sin((2+1)ϕ)se2m+1(t,h2)dt=(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)dt=(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)meν2m1(t,h2)sinh2z+sin2tdt=(1)m+1ihαν,m(0)D0(ν,ν+2m+1,z),
28.28.26 coshzπ202πsintmeν(t,h2)meν2m1(t,h2)sinh2z+sin2tdt=(1)m+1ihαν,m(1)D0(ν,ν+2m+1,z),

where

28.28.27 αν,m(0)=12π02πcostmeν(t,h2)meν2m1(t,h2)dt=(1)m2iπmeν(0,h2)meν2m1(0,h2)hD0(ν,ν+2m+1,0),
28.28.28 αν,m(1)=12π02πsintmeν(t,h2)meν2m1(t,h2)dt=(1)m+12iπmeν(0,h2)meν2m1(0,h2)hD1(ν,ν+2m+1,0).
28.28.29 coshzπ202πsintmeν(t,h2)meν2m1(t,h2)sinh2z+sin2tdt=(1)m+1ihαν,m(0)D1(ν,ν+2m+1,z),
28.28.30 sinhzπ202πcostmeν(t,h2)meν2m1(t,h2)sinh2z+sin2tdt=(1)mihαν,m(1)D1(ν,ν+2m+1,z),
28.28.31 2π202πcostsintmeν(t,h2)meν2m(t,h2)sinh2z+sin2tdt=(1)miγν,mD0(ν,ν+2m,z),
28.28.32 sinh(2z)π202πmeν(t,h2)meν2m(t,h2)sinh2z+sin2tdt=(1)m+1iγν,mD1(ν,ν+2m,z),

where

28.28.33 γν,m=12π02πmeν(t)meν2m(t)dt=(1)m4iπmeν(0)meν2m(0)D1(ν,ν+2m,0).

Also,

28.28.34 1π202πmeν(t,h2)meν2m1(t,h2)sintdt=(1)m+1ihαν,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+sin2tdt=(1)p+1ihα^n,m(s)Ds0(n,m,z),
28.28.37 coshzπ202πsintsen(t,h2)sem(t,h2)sinh2z+sin2tdt=(1)p+1ihα^n,m(s)Ds1(n,m,z),

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

28.28.38 α^n,m(s)=12π02πcostsen(t,h2)sem(t,h2)dt=(1)p2iπsen(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+sin2tdt=(1)p+1ihβ^n,mDsc0(n,m,z),
28.28.42 sinhzπ202πcostsen(t,h2)cem(t,h2)sinh2z+sin2tdt=(1)pihβ^n,mDsc1(n,m,z),

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

28.28.43 β^n,m=12π02πsintsen(t,h2)cem(t,h2)dt=(1)p2iπsen(0,h2)cem(0,h2)hDsc1(n,m,0).

Next,

28.28.44 1π202πsin(2t)sen(t,h2)cem(t,h2)sinh2z+sin2tdt=(1)piγ^n,mDsc0(n,m,z),
28.28.45 sinh(2z)π202πsen(t,h2)cem(t,h2)sinh2z+sin2tdt=(1)p+1iγ^n,mDsc1(n,m,z),

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

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

Lastly,

28.28.47 sinhzπ202πcostcen(t,h2)cem(t,h2)sinh2z+sin2tdt=(1)p+1ihα^n,m(c)Dc0(n,m,z),
28.28.48 coshzπ202πsintcen(t,h2)cem(t,h2)sinh2z+sin2tdt=(1)p+1ihα^n,m(c)Dc1(n,m,z),

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

28.28.49 α^n,m(c)=12π02πcostcen(t,h2)cem(t,h2)dt=(1)p+12iπcen(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).