Special Functions - Sharifsharif.edu/~aborji/25120/files/special functions.pdfAmir Borji Special...

Special Functions

Amir Borji

Contents

1 Bessel, Neumann, and Hankel Functions: Jν(x), Nν(x), H(1)ν (x), H

(2)ν (x) 2

1.1 Analytic Properties and Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Asymptotic Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Small Argument Limit |x| −→ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Large Argument Limit |x| À |ν| , −π < arg(x) < π . . . . . . . . . . . . . . . . . . . . . 31.2.3 Large Order Limit ν −→∞ (positive real values) . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Wronskian relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Integral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Orthogonality Relationships and Fourier-Bessel Series . . . . . . . . . . . . . . . . . . . . . . . 41.5 Semi-Infinite Interval: Fourier-Bessel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Recursion Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.7 Series and Integral Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.8 Far Field Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Modified Bessel Functions Iν(x) and Kν(x) 72.1 Small and Large Argument Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Small Argument Limit |x| −→ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Large Argument Limit |x| −→ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.3 Wronskian relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Integral Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Recursion Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Series Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Spherical Bessel and Hankel Functions: jn(x), nn(x), h(1)n (x), h

(2)n (x) 10

3.1 Asymptotic Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.1 Small Argument Limit |x| −→ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.1.2 Large Argument Limit |x| −→ ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.3 Wronskian and cross products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 Recursion Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Addition Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Orthogonality and Integral Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 Spherical Fourier-Bessel Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Schelkunoff Spherical Bessel Functions: Jn(x), Nn(x), H(1)n (x), H

(2)n (x) 12

4.1 Large Argument Limit |x| À 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Wronskian and cross products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Legendre and Associated Legendre Functions: Pν(z), Qν(z), Pµν (z), Qµ

ν (z) 145.1 Relations Between Associated Legendre Functions . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3 Specific Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.4 Special Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.5 Functional and Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.5.1 Wronskians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1

Amir Borji Special Functions 2

5.6 Orthogonalities and Integral Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.7 Series Relations and Addition Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.7.1 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Gamma Function 20

7 Hypergeometric Function 21

1 Bessel, Neumann, and Hankel Functions: Jν(x), Nν(x), H(1)ν (x), H

(2)ν (x)

Bessel functions are solutions of the following differential equation:

x2y′′ + xy′ + (x2 − ν2)y = 0 (1.1)

Any two of the following functions are linearly independent solutions of (1.1)

Jν(x) Nν(x) H(1)ν (x) H(2)

ν (x)

when ν is not an integer, Jν(x) and J−ν(x) are principal solutions (linearly independent) of (1.1). The Neumannfunction Nν(x) is related to Jν and J−ν :

Nν(x) =cos νπ Jν(x)− J−ν(x)

sin νπ(1.2)

Nn(x) = limν→n

cos νπ Jν(x)− J−ν(x)sin νπ

(1.3)

in some books Neumann functions are denoted by Yν(x) instead of Nν(x). Hankel functions of the first andsecond kind are related to Bessel and Neumann functions:

H(1)ν (z) = Jν(z) + jNν(z) = j

e−jνπJν(z)− J−ν(z)sin νπ

(1.4)

H(2)ν (z) = Jν(z)− jNν(z) =

ejνπJν(z)− J−ν(z)j sin νπ

(1.5)

With a variable transformation x = κρ equation (1.1) can be transformed into:

ρ2y′′ + ρy′ + (κ2ρ2 − ν2)y = 0 (1.6)

When ν = n is an integer Jn and J−n are not independent anymore and we have:

J−n(x) = (−1)nJn(x) N−n(x) = (−1)nNn(x) (1.7)

Plots of the first three Bessel and Neumann functions are shown in Fig. 1.1 and Fig. 1.2, respectively.

1.1 Analytic Properties and Continuation

J±ν(z), Nν(z), H(1)ν (z), and H

(2)ν (z) are regular (holomorphic) functions of z throughout the z-plane cut along

the negative real axis, i.e. they have no singularity in z-plane except a branch cut along arg(z) = ±π. For afixed z, they are entire functions of ν, i.e. they are analytic functions of ν throughout the ν-plane. For ν = ±n(integer), Jν(z) is an entire function of z and has no branch point.

Jν(e±jπx) = e±jνπJν(x) Nν(e±jπx) = e∓jνπNν(x)± 2j cos νπ Jν(x)

H(1)−ν (x) = ejνπH

(1)ν (x) H

(2)−ν (x) = e−jνπH

(2)ν (x)

H(1)ν

(ejπx

)= −e−jνπH

(2)ν (x) H

(2)ν

(ejπx

)= ejνπH

(1)ν (x) + 2 cos νπ H

(2)ν (x)

H(1)ν

(e−jπx

)= e−jνπH

(2)ν (x) + 2 cos νπ H

(1)ν (x) H

(2)ν

(e−jπx

)= −ejνπH

(1)ν (x)

(1.8)


−0.6

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14

−0.2

−0.4

0

J1(x)

J2(x)

J0(x)

Figure 1.1: Bessel functions of the first kind

−3

−1

−0.5

0

0.5

1

0 2 4 6 8 10 12 14

−2

−2.5

−1.5

N2(x)N1(x)

N0(x)

Figure 1.2: Bessel functions of the second kind

1.2 Asymptotic Approximations

1.2.1 Small Argument Limit |x| −→ 0

J0(x) ≈ 1− x2

4≈ 1 (1.9)

Jν(x) ≈

(x

2

)ν

Γ(ν + 1)−→ 1

n!

(x

2

)n(1.10)

Nν(x) ≈ − 1π

Γ(ν)(

2x

)ν

−→ −(n− 1)!π

(2x

)n

n 6= 0 (1.11)

N0(x) ≈ 2π

lnγx

2γ = 1.78107241799 . . . Euler’s constant (1.12)

1.2.2 Large Argument Limit |x| À |ν| , −π < arg(x) < π

Jν(x) ≈√

2π x

cos(x− π

4− νπ

2

)(1.13)

Nν(x) ≈√

2π x

sin(x− π

4− νπ

2

)(1.14)

H(1)ν (x) ≈

√2

π xexp

[j(x− π

4− νπ

2

)]= j−ν

√2

jπ xejx (1.15)

H(2)ν (x) ≈

√2

π xexp

[−j

(x− π

4− νπ

2

)]= jν

√2j

π xe−jx (1.16)

Therefore, Jν and Nν are used to represent standing waves while H(1)ν (x) and H

(2)ν (x) are used to represent

incoming and outgoing traveling waves, respectively.

1.2.3 Large Order Limit ν −→∞ (positive real values)

Jν(x) ≈ 1√2πν

(ex

2ν

)ν(1.17)

Nν(x) ≈ −√

2πν

(ex

2ν

)−ν(1.18)

1.2.4 Wronskian relations

The wronskian between two functions is defined by

W{f, g} , f(x)g′(x)− f ′(x)g(x) (1.19)


W{Jν , Nν} = Jν+1Nν − JνNν+1 =2

πx(1.20)

W{Jν , H(2)ν } = Jν+1H

(2)ν − JνH

(2)ν+1 =

2jπx

(1.21)

W{Nν , H(2)ν } = Nν+1H

(2)ν −NνH

(2)ν+1 = − 2

πx(1.22)

W{H(1)ν , H(2)

ν } = H(1)ν+1H

(2)ν −H(1)

ν H(2)ν+1 =

4jπx

(1.23)

1.3 Integral Representations

Jn(x) =e−jn(α+π

2 )

2π

∫ 2π

0ejx cos(φ−α)ejnφdφ =

12π

∫ 2π

0ejx sin φe−jnφdφ (1.24)

Jn(x) =e−jn(α+π

2 )

2π

∫ π

−πejx cos(φ−α)ejnφdφ =

12π

∫ π

−πejx sin φe−jnφdφ (1.25)

Jn(x) =e−j nπ

2

π

∫ π

0cos(nφ)ejx cos φdφ (1.26)

Jn(x) =1π

∫ π

0cos (x sin(φ)− nφ) dφ (1.27)

J0(x) =2π

∫ π2

0cos (x sin(α)) dα =

2π

∫ π2

0cos (x cos(α)) dα (1.28)

J0(x) =1π

∫ π

0cos (x sin(α)) dα =

1π

∫ π

0cos (x cos(α)) dα (1.29)

Jν(x) =1π

∫ π

0cos (x sin(φ)− νφ) dφ− sin νπ

π

∫ ∞

0e−x sinh t−νtdt <{x} > 0 (1.30)

Jν(x) =2π

∫ ∞

0sin

(x cosh t− νπ

2

)cosh νt dt (1.31)

Nν(x) =1π

∫ π

0sin (x sin(φ)− νφ) dφ− 1

π

∫ ∞

0

(eνt + e−νt cos νπ

)e−x sinh tdt <{x} > 0 (1.32)

Nν(x) = − 2π

∫ ∞

0cos

(x cosh t− νπ

2

)cosh νt dt − 1 < <{ν} < 1 x > 0 (1.33)

1.4 Orthogonality Relationships and Fourier-Bessel Series

Bessel equation (1.6) can be written in the following form:

−(ρy′)′ +ν2

ρy − κ2ρy = 0 =⇒ L[y] = κ2 y L = −1

ρ

∂

∂ρ

(ρ

∂

∂ρ

)+

ν2

ρ2(1.34)

This is a Sturm-Liouville equation with p(ρ) = ρ, q(ρ) = n2

ρ , w(ρ) = ρ, and λ = κ2. With appropriate boundaryconditions on a finite interval such as ρ ∈ [a, b] we can have a Sturm-Liouville eigenvalue problem (regular ifa > 0 and irregular if a = 0). Here n can be any real and non-negative number.

Case I: Consider (1.34) on the interval 0 ≤ ρ ≤ b with boundary condition y(b) = 0. At ρ → 0 we requirethe solution to be bounded. Thus, we can not have Nn(κρ) and eigenfunctions must be in the form of Jn(κρ).Eigenvalues are:

Jn(κb) = 0 =⇒ κm =νnm

b=⇒ λm = (κm)2 =

(νnm

b

)2(1.35)

in which νnm is the mth root of the Bessel function Jn(x) = 0, i.e. Jn(νnm) = 0. The following orthogonalityproperty exists: ∫ b

0Jn

(νnm

bρ)

Jn

(νnk

bρ)

ρ dρ =b2

2[Jn+1(νnm)]2 δmk (1.36)


For any piecewise continuous function f(ρ) for which f(b) = 0 we have:

f(ρ) =∞∑

m=1

FmJn

(νnm

bρ)

(1.37)

in which the coefficients Fm are obtained by using the orthogonality property (1.36)

Fm =2

b2 [Jn+1(νnm)]2

∫ b

0f(ρ)Jn

(νnm

bρ)

ρ dρ (1.38)

Expression (1.37) is called the Fourier-Bessel Series expansion of f(ρ). In particular for the Green’s functionproblem (L− κ2)G(ρ, ρ′) = δ(ρ− ρ′):

δ(ρ− ρ′)ρ′

=∞∑

m=1

Jn

(νnm

bρ)

Jn

(νnm

bρ′

)

b2

2 [Jn+1(νnm)]2(1.39)

G(ρ, ρ′) =∞∑

m=1

Jn

(νnm

bρ)

Jn

(νnm

bρ′

)

b2

2 [Jn+1(νnm)]2[(

νnmb

)2 − κ2] (1.40)

Case II: Consider (1.34) on the interval 0 ≤ ρ ≤ b with boundary condition y′(b) = 0. At ρ → 0 we requirethe solution to be bounded. Again eigenfunctions must be in the form of Jn(κρ) and since y′(b) = 0 we obtainthe eigenvalues:

J ′n(κb) =dJn(κρ)

dρ

∣∣∣ρ=b

= 0 =⇒ κm =ν ′nm

b=⇒ λm = (κm)2 =

(ν ′nm

b

)2

(1.41)

in which ν ′nm is the mth root of the derivative of Bessel function J ′n(x) = 0, i.e. J ′n(ν ′nm) = 0. The followingorthogonality property holds:

∫ b

0Jn

(ν ′nm

bρ

)Jn

(ν ′nk

bρ

)ρ dρ =

b2

2

(1− n2

ν ′2nm

) [Jn(ν ′nm)

]2δmk (1.42)

For any piecewise continuous function f(ρ) for which f ′(b) = 0 we can write:

f(ρ) =∞∑

m=1

FmJn

(ν ′nm

bρ

)(1.43)

in which the coefficients Fm are obtained by using the orthogonality property (1.42)

Fm =2

b2

(1− n2

ν ′2nm

)[Jn(ν ′nm)]2

∫ b

0f(ρ)Jn

(ν ′nm

bρ

)ρ dρ (1.44)

If the interval is [a, b] and a > 0, then the general form of eigenfunctions would be AmJn(κmρ) + BmNn(κmρ).The boundary conditions at ρ = a and ρ = b will determine the eigenvalues κm and we have similar orthogonalityproperty between the eigenfunctions as well.

1.5 Semi-Infinite Interval: Fourier-Bessel Transform

Consider (L− κ2)y = f(ρ) and the corresponding Green’s function problem (L− κ2)G(ρ, ρ′) = δ(ρ−ρ′)ρ over the

interval 0 ≤ ρ < ∞ in which operator L is defined in (1.34) (with Re{n} ≥ 0). Here we require the solution tobe bounded at ρ → 0 and vanish at ρ →∞, then

δ(ρ− ρ′)ρ′

=∫ ∞

0Jn(βρ)Jn(βρ′)β dβ (1.45)


and the Fourier-Bessel Transform pairs are defined by

F (β) =∫ ∞

0f(ρ)Jn(βρ)ρ dρ (1.46)

f(ρ) =∫ ∞

0F (β)Jn(βρ)β dβ (1.47)

The solution to the above boundary value problem is:

G(ρ, ρ′) =∫ ∞

0

Jn(βρ)Jn(βρ′)β2 − κ2

β dβ (1.48)

y(ρ) =∫ ∞

0G(ρ, ρ′)f(ρ′)ρ′ dρ′ =

∫ ∞

0

F (β)β2 − κ2

Jn(βρ)β dβ (1.49)

1.6 Recursion Relationships

Consider Zν(x) to be Jν(x) or Nν(x) or H(1)ν (x) or H

(2)ν (x) or any linear combination of these functions. Then,

the following recursive formulas are applicable (ν can be any number):

Zν−1(x) + Zν+1(x) =2ν

xZν(x) (1.50)

Zν−1(x)− Zν+1(x) = 2Z ′ν(x) (1.51)

Z ′ν(x) +ν

xZν(x) = Zν−1(x) (1.52)

Z ′ν(x)− ν

xZν(x) = −Zν+1(x) (1.53)

[xνZν(x)]′ = xνZν−1(x) (1.54)[x−νZν(x)

]′ = −x−νZν+1(x) (1.55)

in particular Z ′0(x) = −Z1(x). Equation (1.54) and (1.55) are very useful when integrating over Bessel functions.

1.7 Series and Integral Relationships

exp[x

2

(t− 1

t

)]=

+∞∑n=−∞

Jn(x) tn t 6= 0 (1.56)

e−jkρ cos φ =+∞∑

n=−∞(−j)nJn(kρ)ejnφ ejkρ cos φ =

+∞∑n=−∞

jnJn(kρ)ejnφ (1.57)

e−jkρ sin φ =+∞∑

n=−∞(−1)nJn(kρ)ejnφ ejkρ sin φ =

+∞∑n=−∞

Jn(kρ)ejnφ (1.58)

Addition theorems for Bessel and Hankel functions are (all parameters are shown in Fig. 1.3):

Jν(kR)ejνθ =+∞∑

m=−∞Jm(kρ′)Jν+m(kρ)ejm(φ−φ′) ejνφ (1.59)

H(2)ν (kR) ejνθ =

+∞∑m=−∞

Jm(kρ′)H(2)ν+m (kρ) ejm(φ−φ′) ejνφ ρ > ρ′ (1.60)

=+∞∑

m=−∞H(2)

m

(kρ′

)Jν+m(kρ)ejm(φ−φ′) ejνφ ρ < ρ′ (1.61)


R = ρ− ρ′

φ′

θ

ρ′

ρ

φ x

y

Figure 1.3: Parameters for addition theorem

In the following expressions Zn(x) and Bn(x) can be any of Jn(x), Nn(x), H(1)n (x), H

(2)n (x) or linear

combinations of them. m and n are any real numbers.∫

Zn(αx)Bn(βx)x dx = xβZn(αx)Bn−1(βx)− αZn−1(αx)Bn(βx)

α2 − β2(1.62)

= xαZn+1(αx)Bn(βx)− βZn(αx)Bn+1(βx)

α2 − β2(1.63)

∫Z2

n(αx)x dx =x2

2[Z2

n(αx)− Zn−1(αx)Zn+1(αx)]

(1.64)∫

xn+1Zn(x) dx = xn+1Zn+1(x) (1.65)∫

x−n+1Zn(x) dx = −x−n+1Zn+1(x) (1.66)∫

1x

Zn(αx)Bm(αx)dx = αxZn(αx)Bm+1(αx)− Zn+1(αx)Bm(αx)

n2 −m2+

Zn(αx)Bm(αx)n + m

(1.67)∫

Z1(x)dx = −Z0(x) (1.68)∫

xZ0(x)dx = xZ1(x) (1.69)∫ ∞

0e−jkxJ0(kρ)dk =

1√ρ2 − x2

(1.70)

1.8 Far Field Integral

Using stationary phase method or steepest descent technique the following relationship can be derived for farfield r −→∞ region (θ 6= 0, π):

∫ +∞

−∞f(β)H(2)

ν

(ρ√

k2 − β2)

e−jβ(z−z′)dβr→∞−−−→ 2jν+1 e−jkr

rejkz′ cos θf(k cos θ) 0 < θ < π (1.71)

in which f(β) is usually a function of ν as well as β. Here ν can be non-integer. Above equation is very usefulin calculating the far field of antennas in cylindrical structures. For derivative of Hankel function we will obtain

∫ +∞

−∞f(β)H ′(2)

ν

(ρ√

k2 − β2)

e−jβ(z−z′)dβr→∞−−−→ 2jν e−jkr

rejkz′ cos θf(k cos θ) (1.72)

2 Modified Bessel Functions Iν(x) and Kν(x)

Modified Bessel functions are solutions of the following differential equation:

x2y′′ + xy′ − (x2 + ν2)y = 0 (2.1)


which is called the modified Bessel’s differential equation. The general solution of (2.1) can be written as alinear combination of the modified Bessel functions of the first and second kind:

AIν(x) + BKν(x)

When ν is not an integer (ν 6= n) Iν and I−ν are linearly independent (principal) solutions of (2.1), however,we usually use Iν(x) and Kν(x) (also called Kelvin function) which is related to Iν and I−ν :

Kν(x) =π

2 sin νπ[I−ν(x)− Iν(x)] (2.2)

Modified Bessel functions are related to Bessel functions via

Iν(x) = e−j νπ2 Jν(jx) − π < arg(x) ≤ π/2 (2.3)

Iν(x) = ej 3νπ2 Jν(e−j 3π

2 x) π/2 < arg(x) ≤ π (2.4)

Kν(x) = jπ

2ej νπ

2 H(1)ν (jx) − π < arg(x) ≤ π/2 (2.5)

Kν(x) = −jπ

2e−j νπ

2 H(2)ν (−jx) − π/2 < arg(x) ≤ π (2.6)

With a variable transformation x = κρ equation (2.1) can be transformed into:

ρ2y′′ + ρy′ − (κ2ρ2 + ν2)y = 0 (2.7)

whose independent solutions are Iν(κρ) and Kν(κρ). When ν is an integer In and I−n are not independentanymore and we have:

I−n(x) = In(x) K−ν(x) = Kν(x)Iν(e±jπx) = e±jνπIν(x) Kν(e±jπx) = ∓jπ Iν(x) + e∓jνπKν(x)

(2.8)

Plots of the first three modified Bessel functions of the first and second kind are shown in Fig. 2.1 and Fig. 2.2,respectively.

0

8

10

12

0 0.5 1 1.5 2 2.5 3 3.5 4

4

2

6I0(x)

I2(x)

I1(x)

Figure 2.1: Modified Bessel functions of the first kind

0

4

5

0 0.5 1 1.5 2 2.5 3 3.5 4

2

1

3K1(x)

K2(x)

K0(x)

Figure 2.2: Modified Bessel functions of the second kind

2.1 Small and Large Argument Approximations


I0 ≈ 1 +x2

4≈ 1 (2.9)

Iν(x) ≈

(x

2

)ν

Γ(ν + 1)−→ 1

n!

(x

2

)n(2.10)

Kν(x) ≈ Γ(ν)2

(2x

)ν

−→ (n− 1)!2

(2x

)n

ν 6= 0 (2.11)

K0(x) ≈ − lnγx

2γ = 1.78107241799 . . . Euler’s constant (2.12)


2.1.2 Large Argument Limit |x| −→ ∞

Iν(x) ≈ 1√2πx

ex (2.13)

Kν(x) ≈√

π

2xe−x (2.14)

2.1.3 Wronskian relations


W{f, g} , f(x)g′(x)− f ′(x)g(x) (2.15)

W{Kν , Iν} = Kν+1Iν + KνIν+1 =1x

(2.16)

2.2 Integral Representations

I0(x) =1π

∫ π

0e±x cos φ dφ =

1π

∫ π

0cosh [x cosφ] dφ (2.17)

In(x) =1π

∫ π

0ex cos φ cosnφdφ (2.18)

Iν(x) =1π

∫ π

0ex cos φ cos νφ dφ− sin νπ

π

∫ ∞

0e−x cosh t−νtdt (2.19)

K0(x) =∫ ∞

0cos (x sinh t) dt =

∫ ∞

0

cosxt√t2 + 1

dt x > 0 (2.20)

Kν(x) =1

cos νπ2

∫ ∞

0cos (x sinh t) cosh νt dt x > 0 − 1 < <{ν} < 1 (2.21)


Consider Cν(x) to be either of Iν(x) or ejνπKν(x) or any linear combination of these two functions. Then, thefollowing recursive formulas are applicable:

Cν−1(x)− Cν+1(x) =2ν

xCν(x) (2.22)

Cν−1(x) + Cν+1(x) = 2C ′ν(x) (2.23)

C ′ν(x) +

ν

xCν(x) = Cν−1(x) (2.24)

C ′ν(x)− ν

xCν(x) = Cν+1(x) (2.25)

(xνCν(x))′ = xνCν−1(x) (2.26)(x−νCν(x)

)′ = x−νCν+1(x) (2.27)

in particular I ′0(x) = I1(x) and K ′0(x) = −K1(x).

2.4 Series Relationships

exp[x

2

(t +

1t

)]=

+∞∑n=−∞

In(x) tn t 6= 0 (2.28)

ex cos φ =+∞∑

n=−∞In(x)einφ (2.29)


3 Spherical Bessel and Hankel Functions: jn(x), nn(x), h(1)n (x), h

(2)n (x)

Spherical Bessel functions are solutions of the following differential equation:

x2y′′ + 2xy′ + [x2 − n(n + 1)]y = 0 n = 0,±1,±2,±3, . . . (3.1)


jn(x) nn(x) h(1)n (x) h(2)

n (x)

where

jn(x) =√

π

2xJn+ 1

2(x) nn(x) =

√π

2xNn+ 1

2(x)

h(1)n (x) =

√π

2xH

(1)

n+ 12

(x) h(2)n (x) =

√π

2xH

(2)

n+ 12

(x)(3.2)

With a variable transformation x = kr equation (3.1) can be transformed into:

r2y′′ + 2ry′ + [k2r2 − n(n + 1)]y = 0 (3.3)

whose independent solutions are jn(kr), nn(kr) or h(1)n (kr), h

(2)n (kr). Some of the spherical Bessel functions

are listed in the following table.

n jn(x) nn(x) h(1)n (x) h

(2)n (x)

0sinx

x−cosx

x

ejx

jx

e−jx

−jx

1sinx

x2− cosx

x−cosx

x2− sinx

x

(1jx− 1

)ejx

x−

(1jx

+ 1)

e−jx

x

2(

3x3− 1

x

)sinx− 3

x2cosx

(− 3

x3+

1x

)cosx− 3

x2sinx

(3x2− 1 +

3jx

)ejx

jx

(3x2− 1− 3

jx

)e−jx

−jx

Other useful relationships are:

nn(x) = (−1)n+1j−n−1(x) n = 0,±1,±2, · · ·

h(1)−n−1 (x) = j(−1)nh

(1)n (x) h

(2)−n−1 (x) = −j(−1)nh

(2)n (x)

jn(e±jπx) = e±jnπjn(x) nn(e±jπx) = −e±jnπnn(x)

h(1)n

(e±jπx

)= (−1)nh

(2)n (x) h

(2)n

(e±jπx

)= (−1)nh

(1)n (x)

(3.4)

3.1 Asymptotic Approximations


jn(x) ≈ n!(2n + 1)!

(2x)n =xn

1 · 3 · 5 · 7 · · · (2n + 1)(3.5)

nn(x) ≈ −2(2n)!n!

1(2x)n+1

= −1 · 3 · 5 · 7 · · · (2n− 1)xn+1

n = 0, 1, 2, 3, · · · (3.6)


3.1.2 Large Argument Limit |x| −→ ∞

jn(x) ≈ 1x

cos(x− (n + 1)

π

2

)(3.7)

nn(x) ≈ 1x

sin(x− (n + 1)

π

2

)(3.8)

h(1)n (x) ≈ (−j)n+1 ejx

x(3.9)

h(2)n (x) ≈ jn+1 e−jx

x(3.10)

3.1.3 Wronskian and cross products


W{f, g} , f(x)g′(x)− f ′(x)g(x) (3.11)

W{jn, nn} = jnnn−1 − jn−1nn =1x2

(3.12)

W{jn, h(2)n } =

1jx2

(3.13)

W{h(1)n , h(2)

n } =2

jx2(3.14)

jn+1nn−1 − jn−1nn+1 =2n + 1

x3(3.15)


Consider bn(x) to be jn(x) or nn(x) or h(1)n (x) or h

(2)n (x) or any linear combination of these functions. Then,

the following recursive formulas are applicable:

bn−1(x) + bn+1(x) =2n + 1

xbn(x) (3.16)

nbn−1(x)− (n + 1)bn+1(x) = (2n + 1)b′n(x) (3.17)

b′n(x) +n + 1

xbn(x) = bn−1(x) (3.18)

b′n(x)− n

xbn(x) = −bn+1(x) (3.19)

[xn+1bn(x)

]′ = xn+1bn−1(x) (3.20)[x−nbn(x)

]′ = −x−nbn+1(x) (3.21)

in particular b′0(x) = −b1(x).

3.3 Addition Theorems

For R = |r − r′| =√

r2 + r′2 − 2rr′ cos γ with k being an arbitrary complex constant (γ is the angle betweenr and r′):

e−jkR

−jkR=

∞∑

n=0

(2n + 1)jn(kr)h(2)n

(kr′

)Pn(cos γ) r < r′ (3.22)

=∞∑

n=0

(2n + 1)jn(kr′)h(2)n (kr)Pn(cos γ) r > r′ (3.23)

ejkr cos θ =∞∑

n=0

(2n + 1)jn jn(kr)Pn(cos θ) (3.24)


3.4 Orthogonality and Integral Relationships

∫ b

0jn

(νnm

br)

jn

(νnk

br)

r2 dr =b3

2[jn+1(νnm)]2 δmk jn(νnm) = 0 (3.25)

∫ ∞

−∞jn(x)jm(x)dx =

π

2n + 1δnm n,m ≥ 0 (3.26)

jn(x) =(−j)n

2

∫ π

0ejx cos θPn(cos θ) sin θ dθ (3.27)

jn(x) =xn

2n+1n!

∫ π

0cos(x cos θ)(sin θ)2n+1dθ (3.28)

∫b2n(αx)x2 dx =

x3

2[b2n(αx)− bn−1(αx)bn+1(αx)

](3.29)

3.5 Spherical Fourier-Bessel Transform

Spherical Bessel equation (3.3) can be written in the following form:

− 1r2

d

dr

(r2 dy

dr

)+

n(n + 1)r2

y − k2y = 0 =⇒ L[y] = k2 y L = − 1r2

∂

∂r

(r2 ∂

∂r

)+

n(n + 1)r2

(3.30)

This is a Sturm-Liouville equation with p(r) = r2, q(r) = n(n + 1), w(r) = r2, and λ = k2. Consider(L − k2)y = f(r) and the corresponding Green’s function problem (L − k2)G(r, r′) = δ(r−r′)

r2 over the interval0 ≤ r < ∞. Here we require the solution to be bounded at r → 0 and vanish at r →∞, then

δ(r − r′)r2

=2π

∫ ∞

0jn(βr)jn(βr′)β2 dβ (3.31)

and the Spherical Fourier-Bessel Transform pairs are defined by

F (β) =∫ ∞

0f(r)jn(βr)r2 dr (3.32)

f(r) =2π

∫ ∞

0F (β)jn(βr)β2 dβ (3.33)

The solution to the above boundary value problem is:

G(r, r′) =2π

∫ ∞

0

jn(βr)jn(βr′)β2 − k2

β2 dβ (3.34)

4 Schelkunoff Spherical Bessel Functions: Jn(x), Nn(x), H(1)n (x), H

(2)n (x)

Schelkunoff Spherical Bessel functions are solutions of the following differential equation:

y′′ +(

1− n(n + 1)x2

)y = 0 n = 0,±1,±2,±3, . . . (4.1)


Jn(x) Nn(x) H(1)n (x) H(2)

n (x)

whereJn(x) = xjn(x) Nn(x) = xnn(x)

H(1)n (x) = xh

(1)n (x) H

(2)n (x) = xh

(2)n (x)

(4.2)

With a variable transformation x = kr equation (4.1) can be transformed into:

y′′ +(

k2 − n(n + 1)r2

)y = 0 (4.3)

whose independent solutions are Jn(kr), Nn(kr) or H(1)n (kr), H

(2)n (kr). Some of the Schelkunoff spherical

Bessel functions are listed in the following table.


n Jn(x) Nn(x) H(1)n (x) H

(2)n (x)

0 sinx − cosx −jejx je−jx

1sinx

x− cosx −cosx

x− sinx

(1jx− 1

)ejx −

(1jx

+ 1)

e−jx

2(

3x2− 1

)sinx− 3

xcosx

(− 3

x2+ 1

)cosx− 3

xsinx

(3

jx2+ j − 3

x

)ejx

(3j

x2− j − 3

x

)e−jx

Useful relationships for Schelkunoff spherical Bessel functions can be easily derived from those of the sphericalBessel function considering the simple relationship B(x) = xbn(x). Therefore, we will not repeat all of themhere. The most important recursion relations are:

2n + 1x

Bn(x) = Bn−1(x) + Bn+1(x) (4.4)

B′n(x) = Bn−1(x)− n

xBn(x) = Bn+1(x) +

n + 1x

Bn(x) (4.5)

4.1 Large Argument Limit |x| À 1

Jn(x) ≈ cos(x− (n + 1)

π

2

)(4.6)

Nn(x) ≈ sin(x− (n + 1)

π

2

)(4.7)

H(1)n (x) ≈ (−j)n+1ejx (4.8)

H(2)n (x) ≈ jn+1e−jx (4.9)

4.2 Wronskian and cross productsW{Jn, Nn} = JnNn−1 − Jn−1Nn = 1 (4.10)

W{Jn, H(2)n } = −j (4.11)

W{H(1)n , H(2)

n } = −2j (4.12)

Jn+1Nn−1 − Jn−1Nn+1 =2n + 1

x(4.13)


5 Legendre and Associated Legendre Functions: Pν(z), Qν(z), P µν (z), Qµ

ν(z)

In this section z = x+jy and x, y are real. x always means a real number in the interval −1 ≤ x ≤ 1 orx = cos with 0 ≤ θ ≤ π. Also m and n represent non-negative integers m = 0, 1, 2, 3, · · · and n = 0, 1, 2, 3, · · ·

Associated Legendre functions are solutions of the following differential equation:

(1− z2)y′′ − 2zy′ +[ν(ν + 1)− µ2

1− z2

]y = 0 (5.1)

This is a Sturm-Liouville equation with p(z) = 1 − z2, q(z) = µ2

1−z2 , and w(z) = 1. Only singularities of thisequation are ordinary branch points at z = ±1,∞. The branch cut is usually taken on the real axis with−∞ < x < 1. Note that Pµ

ν (x + j0) is different than Pµν (x− j0) for x < 1.

Any of the following functions are solutions to the Associated Legendre differential equation (5.1):

P±µν (±z) P±µ

−ν−1(±z) Q±µν (±z) Q±µ

−ν−1(±z)

Pµν (z) and Qµ

ν (x) are called the Associated Legendre functions of the first and second kind with degree νand order µ, respectively. Above solutions are not always linearly independent. Some of them are linearlyindependent for certain values of ν and µ, some are always linearly independent and some are always linearlydependent. Note that functions Qµ

ν (z) are not defined when ν + µ = −1,−2, −3, · · · . When µ = 0,we have Legendre equation whose solutions are P 0

ν = Pν(±z) and Q0ν = Qν(±z).

5.1 Relations Between Associated Legendre Functions

For complex argument z = x + jy:

Pµ−ν−1(z) = Pµ

ν (z) (5.2)

Qµ−ν−1(z) =

Qµν (z) sin[(ν + µ)π]− πejµπ cos νπ Pµ

ν (z)sin[(ν − µ)π]

(5.3)

Pµν (−z) = e∓jνπPµ

ν (z)− 2π

e−jµπ sin[(ν + µ)π]Qµν (z) (5.4)

Qµν (−z) = −e±jνπQµ

ν (z) (5.5)

P−µν (z) =

Γ(ν − µ + 1)Γ(ν + µ + 1)

[Pµ

ν (z)− 2π

e−jµπ sin(µπ)Qµν (z)

](5.6)

Q−µν (z) = e−j2µπ Γ(ν − µ + 1)

Γ(ν + µ + 1)Qµ

ν (z) (5.7)

In (5.4) and (5.5) the upper sign is for Im[z] > 0 and the lower sign for Im[z] < 0. For real argument−1 ≤ z = x ≤ +1 we have:

Pµ−ν−1(x) = Pµ

ν (x) (5.8)

Qµ−ν−1(x) =

Qµν (x) sin[(ν + µ)π]− π cosµπ cos νπ Pµ

ν (x)sin[(ν − µ)π]

(5.9)

Pµν (−x) = cos[(ν + µ)π]Pµ

ν (x)− 2π

sin[(ν + µ)π]Qµν (x) (5.10)

Qµν (−x) = − cos[(ν + µ)π]Qµ

ν (x)− π

2sin[(ν + µ)π]Pµ

ν (x) (5.11)

P−µν (x) =

Γ(ν − µ + 1)Γ(ν + µ + 1)

[cosµπPµ

ν (x)− 2π

sinµπQµν (x)

](5.12)

Q−µν (x) =

Γ(ν − µ + 1)Γ(ν + µ + 1)

[cosµπQµ

ν (x) +π

2sinµπ Pµ

ν (x)]

(5.13)

The values of these functions on real axis for −1 ≤ x ≤ 1 are defined by:

Pµν (x) =

12

[e

jµπ2 Pµ

ν (x + j0) + e−jµπ2 Pµ

ν (x− j0)]

Qµν (x) =

12e−jµπ

[e−

jµπ2 Qµ

ν (x + j0) + ejµπ2 Qµ

ν (x− j0)]


Qµν (x) =

π

2 sin µπ

[Pµ

ν (x) cos µπ − Γ(ν + µ + 1)Γ(ν − µ + 1)

P−µν (x)

]=

π

2 sin[(ν + µ)π][cos[(ν + µ)π] Pµ

ν (x)− Pµν (−x)]

In most electromagnetic problems µ = ±m is an integer (m = 0, 1, 2, 3, · · · ) and z = x = cos θ, therefore:

Pm−ν−1(x) = Pm

ν (x) (5.14)Qm−ν−1(x) = Qm

ν (x)− π cot νπ Pmν (x) (5.15)

Pmν (−x) = (−1)m

[cos νπPm

ν (x)− 2π

sin νπQmν (x)

](5.16)

Qmν (−x) = −(−1)m

[cos νπQm

ν (x) +π

2sin νπPm

ν (x)]

(5.17)

P−mν (x) = (−1)m Γ(ν −m + 1)

Γ(ν + m + 1)Pm

ν (x) (5.18)

Q−mν (x) = (−1)m Γ(ν −m + 1)

Γ(ν + m + 1)Qm

ν (x) (5.19)

Qmν (x) =

π

2 sin(νπ)[cos(νπ) Pm

ν (x)− (−1)mPmν (−x)] (5.20)

and if ν = n is an integer, too:

Pmn (x) = 0 for m > n (5.21)

Pm−n−1(x) = Pm

n (x) (5.22)Pm

n (−x) = (−1)n+mPmn (x) (5.23)

Qmn (−x) = −(−1)n+mQm

n (x) (5.24)

P−mn (x) = (−1)m (n−m)!

(n + m)!Pm

n (x) (5.25)

Q−mn (x) = (−1)m (n−m)!

(n + m)!Qm

n (x) (5.26)

For µ = m = 0, one can use above equations to deduce relationships between Legendre functions Pν(z) andQν(z). Now we can make the following remarks about independent solutions of (5.1):

(i) When ν + µ is not an integer, Pµν (z) and Qµ

ν (z) are always two linearly independent solutions of (5.1).Exception: when both µ and ν are integers see (ii).

(ii) If µ = ±m and ν = n or ν = −n− 1 then Pmn (x) and Qm

n (x) are again linearly independent solutions forn ≥ m only. Note that Pm

n (−x) and P−mn (x) are proportional to Pm

n (x).

(iii) When ν + µ is not an integer, Pµν (z) and Pµ

ν (−z) are two linearly independent solutions of (5.1). Inparticular, when µ = ±m or µ = 0 and ν is not an integer, we can always use Pm

ν (z) and Pmν (−z) as our

principal solutions.

(iii) Pµν (z) and P−µ

ν (z) are two linearly independent solutions if µ is not an integer (even if ν + µ is integer).

In our electromagnetic problems µ is usually an integer µ = ±m = 0,±1,±2, · · · and we may use:

(I) αPmν (x) + βQm

ν (x) for any value of ν including integers (ν = n).

(II) αPmν (x) + βPm

ν (−x) when ν 6= n (ν is not an integer)

(III) αPµν (x) + βP−µ

ν (x) when µ is not an integer or αPµν (x) + βQµ

ν (x) when ν + µ is not an integer.

When µ = 0 the solutions to (5.1) are Legendre functions of the first and second kind: Pν(z) and Qν(z).Again if ν 6= n we may use αPν(z) + βPν(−z) as principal solution or we can always use αPν(z) + βQν(z).

When ν = n is an integer, Pn(x) is always a polynomial of degree n and they are called Legendre polynomials.If we let x = cos θ equation (5.1) can be transformed into:

1sin θ

d

dθ

(sin θ

dY

dθ

)+

(ν(ν + 1)− µ2

sin2 θ

)Y = 0 0 ≤ θ ≤ π (5.27)

in which Y (θ) = y(x). The principal solution of (5.27) can be written as aPµν (cos θ) + bPµ

ν (− cos θ) oraPµ

ν (cos θ) + bQµν (cos θ).


5.2 Singularities

Here we only focus on Pmν (±x) and Qm

ν (x) which often appear in electromagnetic problems (m = 0, 1, 2, 3, · · · ).(a) For non-integer values of ν (ν 6= n), Pm

ν (x) is always unbounded (infinite) at x = −1 and finite (bounded)at x = 1. For m < 0 the same result applies - see (5.18). Therefore, we use only Pm

ν (cos θ) when θ = π isnot in the region but θ = 0 is included. On the other hand, we only use Pm

ν (− cos θ) when θ = 0 is notthe region but θ = π is included.

(b) Pmn (x) (when ν = n is integer) is always bounded at x = ±1. Thus, we can only use Pm

n (cos θ) whenθ = 0, π are both in the region.

(c) From (5.20), Qmν (x) is always singular (unbounded) at x = ±1. Thus, they cannot be used when either

θ = 0 or θ = π are in the region

(d) When neither θ = 0, π are inside the region, we must use a linear combination: APmν (cos θ)+BPm

ν (− cos θ)or APm

ν (cos θ) + BQmν (cos θ)

5.3 Specific Examples

In Fig. 5.1 and Fig. 5.2 several Legendre functions of the first and second kind are shown.

−1−0.5 0 0.5 1

1

0.5

0

−0.5

−1

P3(x)

P4(x)

P2(x)

P1(x)

P0(x)

Figure 5.1: Legendre polynomials Pn(x)

−3

2

3

−1 −0.5 0 0.5 1

0

−1

−2

1 Q3(x)

Q2(x)

Q1(x)

Q0(x)

Figure 5.2: Legendre functions of the second kind Qn(x)

n m Pmn (x) Pm

n (cos θ)

0 0 1 1

1 0 x cos θ

1 1 −√1− x2 − sin θ

2 012

(3x2 − 1

) 14

(3 cos 2θ + 1)

2 1 −3x√

1− x2 −3 sin θ cos θ

2 2 3(1− x2)32(1− cos 2θ)

3 012

(5x3 − 3x

) 18

(5 cos 3θ + 3 cos θ)

3 1 −32

√1− x2(5x2 − 1) −3

8(sin θ + 5 sin 3θ)

3 2 15x(1− x2)154

(cos θ − cos 3θ)

3 3 −15(1− x2)√

1− x2 −154

(3 sin θ − sin 3θ)


n m Qmn (x) Qm

n (cos θ)

0 012

ln1 + x

1− xln

(cot

θ

2

)

1 0x

2ln

1 + x

1− x− 1 cos θ ln

(cot

θ

2

)− 1

1 1 −(1− x2)(

12

ln1 + x

1− x+

x

1− x2

)

2 014

(3x2 − 1

)ln

1 + x

1− x− 3x

2

2 1 −√1− x2

(3x

2ln

1 + x

1− x+

3x2 − 21− x2

)

2 2√

1− x2

(32

ln1 + x

1− x+

5x− 3x3

(1− x2)2

)

5.4 Special Values

Pmn (x) = 0 m > n (5.28)

P−1ν (cos θ) =

−1ν(ν + 1)

dPν(cos θ)dθ

(5.29)

P 1ν (cos θ) =

dPν(cos θ)dθ

(5.30)

P−νν =

1Γ(1 + ν)

(sin θ

2

)ν

(5.31)

Pν(1) = 1 (5.32)

Pν(0) = −12

sin νπ√π3

Γ(

ν + 12

)Γ

(−ν

2

)(5.33)

Pmn (0) =

(−1)n+m

21 · 3 · 5 · · · (n + m− 1)

2 · 4 · 6 · · · (n−m)n + m : even

0 n + m : odd(5.34)

Qmn (0) =

0 n + m : even

(−1)n+m+1

22 · 4 · 6 · · · (n + m− 1)

1 · 3 · 5 · · · (n−m)n + m : odd

(5.35)

dq

dxqPm

n (x)∣∣∣x=0

= (−1)qPm+qn (0) (5.36)

dq

dxqQm

n (x)∣∣∣x=0

= (−1)qQm+qn (0) (5.37)

Pmn (±1) = 0 m > 0 (5.38)Pn(1) = 1 Pn(−1) = (−1)n (5.39)

Pmn (−x) = (−1)n+mPm

n (x) (5.40)

P ′n(1) =

n(n + 1)2

(5.41)

Pµν (0) =

2µ√π

Γ(1 + ν−µ

2

)Γ

(1−ν−µ

2

) (5.42)

dPµν (0)dx

=2µ+1 sin (ν+µ)π

2 Γ(1 + ν+µ

2

)√

πΓ(

1+ν−µ2

) (5.43)


5.5 Functional and Recurrence Relations

Here we only consider real argument −1 ≤ z = x ≤ 1. The following relations are satisfied by both Pµν (x) and

Qµν (x) unless otherwise stated. For µ = 0 one obtains the relations for Legendre functions Pν and Qν :

Pmν (x) = (−1)m

(1− x2

)m2

dmPν(x)dxm

also for Qmν (5.44)

Pn(x) =1

2n n!dn

dxn(x2 − 1)n only for Pn(x) (5.45)

(1− x2)dPµ

ν (x)dx

= (ν + 1)xPµν (x)− (ν + 1− µ)Pµ

ν+1(x) (5.46)

= (ν + µ)Pµν−1(x)− νxPµ

ν (x) (5.47)

= −√

1− x2Pµ+1ν (x)− µxPµ

ν (x) (5.48)

= (ν + 1− µ)(ν + µ)√

1− x2Pµ−1ν (x) + µxPµ

ν (x) (5.49)√

1− x2Pµ−1ν (x) =

Pµν−1(x)− Pµ

ν+1(x)2ν + 1

=Pµ

ν−1(x)− xPµν (x)

ν − µ + 1(5.50)

√1− x2Pµ+1

ν (x) =(ν − µ)(ν − µ + 1)Pµ

ν+1(x)− (ν + µ)(ν + µ + 1)Pµν−1(x)

2ν + 1(5.51)

= (ν − µ)xPµν (x)− (ν + µ)Pµ

ν−1(x) (5.52)(2ν + 1)xPµ

ν (x) = (ν + 1− µ)Pµν+1(x) + (ν + µ)Pµ

ν−1(x) (5.53)√

1− x2dPν(x)

dx= −P 1

ν (x) (5.54)

P ′n+1(x)− P ′

n−1 = (2n + 1)Pn(x) (5.55)

5.5.1 Wronskians W {Pν(x), Qν(x)} =1

1− x2(5.56)

W{P−µ

ν (x), Pµν (x)

}=

2 sin µπ

π(1− x2)(5.57)

W {Pν(x), Pν(−x)} = − 2 sin νπ

π(1− x2)(5.58)

W {Pµν (x), Qµ

ν (x)} =22µ

1− x2

Γ(ν+µ

2 + 12

)Γ

(ν+µ2 + 1

)

Γ(ν−µ

2 + 12

)Γ

(ν−µ2 + 1

) (5.59)

5.6 Orthogonalities and Integral Relationships

In the following δnk and δνα are the Kronecker delta: δnk = 1 if n = k and δnk = 0 if n 6= k. Moreover, Lµν (x)

represents any of Pµν (x), Pµ

ν (−x), or Qµν (x). Also note that m = 0, 1, 2, · · · , n

∫ θ0

0Pν(cos θ) sin θ dθ = sin θ P−1

ν (cos θ) (5.60)∫ θ0

0Pν(cos θ) cos θ sin θ dθ =

− sin θ

(ν − 1)(ν + 2)[sin θ Pν(cos θ) + cos θP 1

ν (cos θ)]

(5.61)∫ 1

−1Pm

n (x)Pmk (x)dx =

∫ π

0Pm

n (cos θ)Pmk (cos θ) sin θ dθ =

22n + 1

(n + m)!(n−m)!

δnk (5.62)

∫ 1

−1Pm

n (x)P−mk (x)dx =

22n + 1

(−1)m δnk (5.63)

∫ 1

−1Pm

n (x)P kn (x)

dx

1− x2=

∫ π

0Pm

n (cos θ)P kn (cos θ)

dθ

sin θ=

1m

(n + m)!(n−m)!

δmk (5.64)

∫ 1

−1xmPn(x) dx = 0 m < n (5.65)


∫ 1

−1xnPn(x) dx =

2n+1(n!)2

(2n + 1)!(5.66)

∫ 1

−1

Pn(x)√1− x

dx =√

82n + 1

(5.67)∫ π

0ejx cos θPn(cos θ) sin θ dθ = 2jn jn(x) (5.68)

∫ π

0cos (x cos θ)Pn(cos θ) sin θ dθ =

{0 n : odd2(−1)

n2 jn(x) n : even

(5.69)

∫ π

0

e−jk√

a2+b2−2ab cos θ

√a2 + b2 − 2ab cos θ

Pn(cos θ) sin θ dθ = −2jk jn(ka)h(2)n (kb) 0 ≤ a ≤ b (5.70)

∫ π

0

(dPm

n

dθ

dPmk

dθ+

m2

sin2 θPm

n (cos θ)Pmk (cos θ)

)sin θ dθ =

2n(n + 1)2n + 1

(n + m)!(n−m)!

δnk (5.71)∫ π

0

(P 1

n(cos θ)sin θ

dP 1k

dθ+

P 1k (cos θ)sin θ

dP 1n

dθ

)sin θ dθ = 0 (5.72)

∫ x2

x1

Lµν (x)Lµ

α(x)dx =∫ θ1

θ2

Lµν (cos θ)Lµ

α(cos θ) sin θ dθ =δνα

2ν + 1

[sin θ

∂Lµν (cos θ)∂θ

∂Lµν (cos θ)∂ν

]θ1

θ2

(5.73)

Above equation is valid if Lµν (cos θ1) = Lµ

ν (cos θ2) = 0 or if θ2 = 0 when Lµν (x) = Pµ

ν (x) or if θ2 = π whenLµ

ν (x) = Pµν (−x).

∫ x2

x1

Lµν (x)Lµ

α(x)dx =∫ θ1

θ2

Lµν (cos θ)Lµ

α(cos θ) sin θ dθ = − δνα

2ν + 1

[sin θLµ

ν (cos θ)∂2Lµ

ν (cos θ)∂θ∂ν

]θ1

θ2

(5.74)

Above equation is valid if ∂Lµν (cos θ1)

∂θ = ∂Lµν (cos θ2)

∂θ = 0 or if θ2 = 0 when Lµν (x) = Pµ

ν (x) or if θ2 = π whenLµ

ν (x) = Pµν (−x).

∫ θ1

θ2

[∂Lµ

ν (cos θ)∂θ

∂Lµα(cos θ)∂θ

+µ2

sin2 θLµ

ν (cos θ)Lµα(cos θ)

]sin θ dθ = δνα

ν(ν + 1)2ν + 1

[sin θ

∂Lµν (cos θ)∂θ

∂Lµν (cos θ)∂ν

]θ1

θ2

(5.75)Above equation is valid if Lµ

ν (cos θ1) = Lµν (cos θ2) = 0 or if θ2 = 0 when Lµ

ν (x) = Pµν (x) or if θ2 = π when

Lµν (x) = Pµ

ν (−x).

∫ θ1

θ2

[∂Lµ

ν (cos θ)∂θ

∂Lµα(cos θ)∂θ

+µ2

sin2 θLµ

ν (cos θ)Lµα(cos θ)

]sin θ dθ = −δνα

ν(ν + 1)2ν + 1

[sin θLµ

ν (cos θ)∂2Lµ

ν (cos θ)∂θ∂ν

]θ1

θ2

(5.76)Above equation is valid if ∂Lµ

ν (cos θ1)∂θ = ∂Lµ

ν (cos θ2)∂θ = 0 or if θ2 = 0 when Lµ

ν (x) = Pµν (x) or if θ2 = π when

Lµν (x) = Pµ

ν (−x).

5.7 Series Relations and Addition Theorems

Here εm is the Neumann number (εm = 1 if m = 0 and εm = 2 if m > 0). Also we have defined

cos γ = cos θ cos θ′ + sin θ sin θ′ cos(φ− φ′) (5.77)

Pn(cos γ) =n∑

m=0

εm(n−m)!(n + m)!

Pmn (cos θ)Pm

n (cos θ′) cos m(φ− φ′) (5.78)

=n∑

m=−n

(n−m)!(n + m)!

Pmn (cos θ)Pm

n (cos θ′)ejm(φ−φ′) (5.79)


Pν(cos γ) =∞∑

m=0

εm(−1)mP−mν (cos θ)Pm

ν (cos θ′) cos m(φ− φ′) (5.80)

0 < θ, θ′ < π 0 < θ + θ′ < π

Qν(cos γ) =∞∑

m=0

εm(−1)mP−mν (cos θ)Qm

ν (cos θ′) cos m(φ− φ′) (5.81)

0 < θ < π/2 0 < θ′ < π 0 < θ + θ′ < π

If we define R = |r − r′| =√

r2 + r′2 − 2rr′ cos γ:

1R

=∞∑

n=0

rn<

rn+1>

Pn(cos γ) (5.82)

e−jkR

−jkR=

∞∑

n=0

(2n + 1)jn(kr<)h(2)n (kr>) Pn(cos γ) =

=∞∑

n=0

(2n + 1)jn(kr<)h(2)n (kr>)

n∑

m=0

εm(n−m)!(n + m)!

Pmn (cos θ)Pm

n (cos θ′) cos m(φ− φ′) (5.83)

(5.84)

5.7.1 Spherical Harmonics

Spherical harmonics denoted by Ynm(θ, φ) are the normalized solutions of

1sin θ

∂

∂θ

(sin θ

∂Y

∂θ

)+

1sin2 θ

∂2Y

∂φ2+ n(n + 1)Y = 0 (5.85)

for which we obtain

Ynm(θ, φ) =

√2n + 1

4π

(n−m)!(n + m)!

Pmn (cos θ)ejmφ (5.86)

with n = 0, 1, 2, 3, · · · and m = 0,±1,±2, · · · ,±n. Sometimes we use

Ynm(θ, φ) =

√2n + 1

4π

(n−m)!(n + m)!

Pmn (cos θ)

{cosmφsinmφ

}(5.87)

with n = 0, 1, 2, 3, · · · and m = 0, 1, 2, · · · , n. These functions are also called Tesseral Harmonics for m < nand Sectoral Harmonics for m = n. These functions are orthonormal over the surface unit sphere. When usingthe definition in (5.86), the following relationships hold:

∫ 2π

0

∫ π

0Ynm(θ, φ)Y ∗

kl(θ, φ) sin θ dθ dφ = δnk δml (5.88)

∞∑

n=0

n∑m=−n

Ynm(θ, φ)Y ∗nm(θ′, φ′) = δ(φ− φ′)δ(θ − θ′) (5.89)

Yn,−m = (−1)mY ∗nm (5.90)

Pn(cos γ) =4π

2n + 1

n∑m=−n

Ynm(θ, φ)Y ∗nm(θ′, φ′) (5.91)

n∑m=−n

|Ynm(θ, φ)|2 =2n + 1

4π(5.92)

6 Gamma Function

Γ(z) =∫ ∞

0e−ttz−1dt = λz

∫ ∞

0e−λttz−1dt Re[z] > 0 Re[λ] > 0 (6.1)


Γ(z) is an analytic function with simple poles at z = −m for m = 0, 1, 2, 3, . . . with the corresponding residue

equal to(−1)m

m!

Γ(x)ejαx = λx

∫ ∞

0tx−1 exp

(−λte−jα)dt x > 0 λ > 0 − π

2< α <

π

2(6.2)

Γ(1 + z) = zΓ(z) (6.3)Γ(n + 1) = n! (6.4)

Γ(1) = Γ(2) = 1 (6.5)

Γ(

12

)=√

π (6.6)

Γ(z)Γ(1− z) =π

sinπz(6.7)

Γ(−z)Γ(z) =−π

z sinπz(6.8)

Γ(

12

+ z

)Γ

(12− z

)=

π

cosπz(6.9)

Γ (1 + z) Γ (1− z) =πz

sinπz(6.10)

Γ(

32

+ z

)Γ

(32− z

)=

π

cosπz

(14− z2

)(6.11)

Γ(2 + z) = z(1 + z)Γ(z) (6.12)

Γ(2z) =22z−1

√π

Γ(z)Γ(

12

+ z

)(6.13)

Γ(jx) =Γ(1 + jx)

jx+ πδ(x) (6.14)

B(x, y) =Γ(x)Γ(y)Γ(x + y)

(6.15)

is called the Beta function.∫ π

2

0sinµ x dx =

∫ π2

0cosµ x dx = 2µ−1B

(µ + 1

2,µ + 1

2

)(6.16)

∫ π2

0sinµ x cosν x dx =

12B

(µ + 1

2,ν + 1

2

)Re[µ] > 0 Re[ν] > 0 (6.17)

∫ π2

0ej(µ+ν)x sinµ−1 x cosν−1 x dx = ejµ π

2 B (µ, ν) Re[µ] > 0 Re[ν] > 0 (6.18)

∫ π2

−π2

e−jλθ cosµ θ(a2ejθ + b2e−jθ

)νdθ =

πb2ν2F1

(−ν,−λ+ν+µ

2 ; 1 + µ−ν−λ2 ; a2

b2

)

2µ(µ + 1)B(1− λ+ν−µ

2 , 1 + λ+ν+µ2

) a2 < b2 (6.19)

=πa2ν

2F1

(−ν, λ−ν−µ

2 ; 1 + µ−ν+λ2 ; b2

a2

)

2µ(µ + 1)B(1 + λ+µ−ν

2 , 1 + −λ+ν+µ2

) b2 < a2 (6.20)

7 Hypergeometric Function

limγ→−n

F (α, β; γ; z)Γ(γ)

=Γ(α + n + 1)Γ(β + n + 1)

Γ(α)Γ(β)zn+1

(n + 1)!F (α + n + 1, β + n + 1;n + 2; z) (7.1)

limε→0

F (α, β; ε; z)Γ(ε)

= αβzF (α + 1, β + 1; 2; z) (7.2)


F (α, β; γ; 0) = 1 (7.3)

F (α, β; γ; 1) =Γ(γ)Γ(γ − α− β)Γ(γ − α)Γ(γ − β)

Re[γ] > Re[α + β] (7.4)

= F (−α,−β; γ − α− β; 1) (7.5)

=1

F (−α, β; γ − α; 1)(7.6)

F (α, β; γ; z) = F (β, α; γ; z) (7.7)

F (α, β; γ; z) = (1− z)−αF (α, γ − β; γ;z

z − 1) (7.8)

= (1− z)γ−α−βF (γ − α, γ − β; γ; z) (7.9)

=Γ(γ)Γ(γ − α− β)Γ(γ − α)Γ(γ − β)

F (α, β; α + β − γ + 1; 1− z)+

(1− z)γ−α−β Γ(γ)Γ(α + β − γ)Γ(α)Γ(β)

F (γ − α, γ − β; γ − α− β + 1; 1− z) (7.10)

= (1− z)−α Γ(γ)Γ(β − α)Γ(β)Γ(γ − α)

F (α, γ − β;α− β + 1;1

1− z)+

(1− z)−β Γ(γ)Γ(α− β)Γ(α)Γ(γ − β)

F (β, γ − α; β − α + 1;1

1− z) (7.11)

∫ ∞

0

Jν(αx)Jµ(βx)xλ

dx =

(α

β

)ν

βλ−1Γ(

ν+µ−λ+12

)

2λΓ(−ν+µ+λ+1

2

)Γ(ν + 1)

F

(ν + µ− λ + 1

2,ν − µ− λ + 1

2; ν + 1;

α2

β2

)(7.12)

For Re[ν + µ− λ + 1] > 0 and Re[λ] > −1 and 0 < α < β

∫ ∞

0

Jν(αx)Jµ(βx)xλ

dx =

(β

α

)µ

αλ−1Γ(

µ+ν−λ+12

)

2λΓ(−µ+ν+λ+1

2

)Γ(µ + 1)

F

(µ + ν − λ + 1

2,µ− ν − λ + 1

2; µ + 1;

β2

α2

)(7.13)

For Re[ν + µ− λ + 1] > 0 and Re[λ] > −1 and 0 < β < α

∫ ∞

0

Jν(αx)Jµ(αx)xλ

dx =αλ−1Γ(λ)Γ

(ν+µ−λ+1

2

)

2λΓ(

µ−ν+λ+12

)Γ

(µ+ν+λ+1

2

)Γ

(ν−µ+λ+1

2

) (7.14)

For Re[ν + µ + 1] > Re[λ] > 0 and α > 0

Special Functions - Sharifsharif.edu/~aborji/25120/files/special functions.pdfAmir Borji Special...

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Transcript of Special Functions - Sharifsharif.edu/~aborji/25120/files/special functions.pdfAmir Borji Special...