Preamble

DIFFERENTIAL EQUATIONS

1967, Vol. III, No. 3
UDC 517.564.4

EXPANSION OF AN ARBITRARY FUNCTION INTO AN INTEGRAL OVER SQUARES OF LEGENDRE FUNCTIONS WITH COMPLEX INDEX

In the study of many problems in mathematical physics, a significant role is played by the representation of an arbitrary function $f(x)$ defined on a given interval as a Mehler–Fock integral:

$$f(x) = \int_{0}^{\infty} \tau \tanh(\pi \tau) P_{-\frac{1}{2}+i\tau}(x) d\tau \int_{1}^{\infty} f(y) P_{-\frac{1}{2}+i\tau}(y) dy$$

This expansion is fundamental for solving boundary value problems in coordinate systems where the boundaries are surfaces of constant values in conical or toroidal coordinates. The Legendre functions of complex index, $P_{-\frac{1}{2}+i\tau}(x)$, form the basis of this integral transform, which generalizes the Fourier transform to cases involving spherical symmetry and hyperbolic geometries.

The validity of such expansions depends on the properties of the function $f(x)$ and the asymptotic behavior of the Legendre functions. In this paper, we investigate the conditions under which an arbitrary function can be decomposed into an integral involving the squares of these Legendre functions. Such representations are particularly useful when dealing with quadratic forms of fields or energy densities in physical applications.

1 < X < o o ,

where $P$ is the spherical Legendre function of the first kind. The objective of the present work is to derive a formula of a similar type, but one that contains the squares of Legendre functions. In this paper, the following theorem is proven.

Let $f(x)$ be an arbitrary function defined on a given interval and satisfying the conditions $f(x) \in L(1, a)$ and $f(x) \in \dots$. Then the following expansion holds:

[FIGURE:1]

$$\begin{aligned} f(x) = \dots \end{aligned}$$

1 < X < 00 ,

$(z)$ is the Legendre function of the first kind. The integrals with respect to the variable in the left and right sides of equation (3) are Lebesgue integrals, while the integral with respect to the variable $\tau$ is understood as the limit of the corresponding Riemann integral over the interval as $T \to \infty$.

This theorem allows us to obtain the inversion of the integral transform

$$F(\tau) = L[x(t)] = \int_{1}^{\infty} x(t) P_{-\frac{1}{2} + i\tau}(t) dt$$

,.(*) . ( * ) ! f(x)dx,

$0 < \tau < \infty$

Under the assumption that $f(x)$ belongs to the class of functions satisfying conditions (2), the inversion formula is given by:
$$f(x) = \frac{1}{2\pi i} \int_{\sigma - i\infty}^{\sigma + i\infty} \frac{\Gamma(s)}{\Gamma(1-s)} x^{-s} F(s) ds \quad (5)$$
holding for almost all $x$. The application of the expansion (3) to functions of a particular form leads to relations that are of interest in the theory of special functions and in mathematical physics.

1. ASYMPTOTIC REPRESENTATIONS OF SPHERICAL FUNCTIONS

The proof of the theorem relies on certain asymptotic representations of Legendre functions for large absolute values of the index. To derive the first of the required representations, we utilize the well-known equality:

$\Gamma(r - \nu)$

( c h a ) =

a > 0, Rev > — 2

and introduce a new integration variable $\theta - \alpha$. We then obtain

$$ e \approx \omega \left( \Gamma \right) \rho^{- \nu} Q, \quad \cosh \alpha = - \frac{\Lambda - \gamma}{\sqrt{2 \sinh \alpha}} \int \sqrt{\sinh \theta} \, d\theta $$

G—V*. " T I E « > •

where $g(t) = 1 - |1 + \coth \alpha - \tanh \gamma|$

The integral $i_R$ can be expressed in terms of gamma functions.

According to Lebedev, it follows that $i = (-\pi n_0 Q \sqrt{1 - |v|^{-2}} \arg v)$. Regarding the integral $I_L$, by integrating by parts, we find:
$$\frac{g'(t)}{g(t) \cosh t / \sinh^{3/2} t}$$
From this, taking into account that $g(t)$ is a positive, monotonically increasing function, we obtain $\frac{g'(t)}{g(t) \cosh t}$.

1^1 <

Substituting $g'(t)$ and evaluating the integral on the right-hand side of the final equation using the substitution $u = \coth(y)$, we find:

$$ \sqrt{\coth(a)} $$

v| J (1 + ^ ) 3 / 2 < ^ c t h a

Consequently,

£ , = / c t h a O f l v l - 1 ) ,

Ivl -> oo, largvl <

From equations (7)–(9), we derive the asymptotic representation:

$$ \Phi(\text{ch} \alpha) = \left( e^{-\alpha} + \sqrt{\text{cth} \alpha} \right) $$

| v | - * o o , | a r g v | < — , a > 0 ,

where the symbol depends on $a$. Deriving an analogous formula for the Legendre function of the second kind is more complex. We proceed from the integral representation:

d 0 =

|/"2ch 2ch 0

Assuming $t = a$ and $Re(\nu) > 0$, we can represent the first of the integrals as a sum:
$$\int_{0}^{2a} \left( \int_{0}^{t} \sinh \tau \, d\tau + \int_{t}^{2a} \sinh \tau \, d\tau \right)^{-1/2}$$

where $h(t) = \left( 1 - \coth a \cdot \tanh \frac{t}{2} \right)^{-1}$.

1. We have

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$|vl \ oo, \argv| < y$

Next, by integrating by parts, we find:

$$\begin{aligned} \dots \end{aligned}$$

from which, by estimating the modulus, we obtain:

$$-f^L- = - ^ U - Oflvl-$$

|v|->oo, | a r g v | < ~ - .

By performing integration by parts on the last integral in (12), we can represent it in the following form:

h(t)dt = J — 7 = = ~ e ~™ —

] / s h a 7 — h'(t)dt h(t)chtdt

Mathematical Analysis

If we observe that $h(t)$ is a positive, monotonically increasing function on the interval $(0, a)$, then the following estimate can be derived from the preceding equality after straightforward transformations:

$$ \frac{h'(t)}{h(t)} \leq \frac{s}{t} \frac{h'(t)}{h(t)} $$

[FIGURE:1]

This relationship characterizes the growth rate of the function within the specified domain. By analyzing the differential properties of $h(t)$, we can establish bounds that are critical for the subsequent convergence proofs. The monotonicity ensures that the derivative $h'(t)$ remains non-negative, which simplifies the application of the mean value theorem in this context.

Furthermore, the behavior of the ratio $\frac{h'(t)}{h(t)}$ provides insight into the logarithmic growth of the system. Such estimates are standard in the study of integral transforms and special functions, particularly when determining the asymptotic behavior of solutions near the boundary of the interval $(0, a)$.

= dt =

] / \cosh a \tanh a / \cosh a - \dots )^{3/2}, from which it follows, as shown above, that

$$h(t) dt = \sqrt{\cosh a} / \cosh a + O(\nu^{-1})$$ (15)

|v|-> oo, | a r g v | < y

It follows from (12)—(15) that

1 5 , =

e w { l - f / c t h a l /chaOdvl- 1 / 2 )} |v| oo, |arg v| <

To obtain the asymptotic representation of the integral, we observe that it can be transformed into the following form:

$$ \int \frac{d\theta}{2 \cosh \theta - 2 \cosh \alpha} $$

[FIGURE:1]

*) Note: The specific limits and coefficients of this transformation depend on the boundary conditions established in the previous section.

6 " = d 0 + o J / 2 c h a + 2che

where the upper or lower sign is chosen depending on the sign of $\text{Im } \nu \gtrless 0$.

ni J j ^2cha — 2cosi|)

Formula (17) for $\text{Im } \nu < 0$ is obtained by integrating the function $e^{\nu \alpha} (2\cosh \alpha - 2\cosh \theta)^{-1/2}$ along a contour formed by the semi-infinite line $(\theta, \theta + i\infty)$, the segment $[-\theta, \theta]$, and the semi-infinite line $(-\theta, -\theta + i\infty)$, while bypassing the branch points. For $\text{Im } \nu > 0$, the chosen contour is the mirror image of the previous contour relative to the real axis.

The asymptotic representation of the first integral in (17) is given by formula (10). By performing integration by parts and estimating the resulting integrals in terms of their moduli, we obtain the asymptotic representations for the remaining two integrals: $O(|\nu|^{-1/2} e^{\text{Re } \nu \alpha + 2\cosh \theta})$.

$|\arg \nu| < \frac{\pi}{2}$

]/ 2ch a —2 cos

$\infty, |\arg v| < \gamma$

From (17)–(19) it follows that $\frac{z \nu}{\sinh a} / \frac{\nu}{\sinh a}$

$|v| \to \infty, |\arg v| < \gamma$.

Formulas (11), (16), and (20) show that the following asymptotic representation holds:

$\left( \frac{1}{\pi} \right)^{1/2} P_1(\cosh a) = \dots$

\2nv-sha / ie~™+ e / c t h a / c h a Odvl"-

$|v| \to \infty, \quad |\arg v| < \gamma, \quad a > 0,$

where the sign $\pm$ is chosen accordingly, and the symbol depends on $a$. The derived asymptotic formulas (10) and (21) differ from conventional asymptotic representations in that they provide an estimate of the remainder terms that remains valid across the entire interval of the variable's variation.

§ 2. PROOF OF THE DECOMPOSITION THEOREM

Proceeding to the proof of equality (3), we first note that from the integral representation for the product of spherical functions:

LEBEDEV

$1 < x < \infty, \tau > 0$,

and the inequality $|(\cdot)| \leq \dots$, the following estimate follows:

$< 2 P_{-\frac{1}{2} + i\tau}(x) Q_{-\frac{1}{2} + i\tau}(x)$,

$1 < x < \infty, \tau > 0$.

It follows that the inner integral on the right-hand side of the formula is majorized by the integral $\int_{1}^{\infty} |Q_{\nu}(y)| |f(y)| dy$, which converges due to the conditions imposed on the function $f(x)$. Furthermore, this integral represents a continuous function on the interval, and the iterated integral is well-defined.

$$J(x, T) = \frac{1}{2} \int_{1}^{x} \frac{1}{\sqrt{x^2 - 1}} \left[ \int_{0}^{T} \tanh(\pi \tau) P_{-\frac{1}{2} + i\tau}(x) d\tau \right] dx$$

The expression $\int [Q_{\nu}(y) + Q_{-\nu}(y)] f(y) dy$ is meaningful. Furthermore, in view of dominated convergence, the order of integration can be interchanged, such that:
$$\int f(y) G(x, y, T) dy = \int \left[ \int (Q_{\nu}(y) + Q_{-\nu}(y)) f(y) dy \right] dT$$

$1 < x < \infty, \quad 1 < y < \infty, \quad T > 0$.

By setting these values and utilizing well-known functional relations, we obtain:

P _ v - i ^ ) = P

Note that $\Gamma \approx 0 \left( \frac{1}{l} \ln(w-1) \right)$.

$-2 \approx y \approx O(\ln \eta)$, where $a < y < \infty$.

By substituting $P(x) = Q_{-\frac{1}{2}}(x) - Q_{\frac{1}{2}}(x)$, we obtain the integral representation:

$G(x, y, T) = \sqrt{T} \sqrt{y} \sqrt{T - x}$

The expression found is convenient for studying the function when $y > x$. To achieve this, we rewrite (28) in the form:

$G(x, y, T) = \sqrt{T-x} \sqrt{y} - \sqrt{y-x}$

$W Q\_W P\_(y) Q(t/) dv$. Taking into account that the integrand in the first integral is an odd function, it follows that this integral is equal to zero. Thus, we obtain:

$G(x, y, T) = \sqrt{y^i}$

$(y)dv$. The functions under the integral signs in (28) and (29) possess no singularities in the half-plane. Consequently, the integration along the segment of the imaginary axis can be replaced by integration along a circular arc of radius lying within this half-plane. Thus, we have:

$G(x, y, T) = \sqrt{x^2 - T} \sqrt{T^2 - |y - x|^2}$

v _ J > ) f [ Q v _ j _ ( < / ) ] 2 < b , (30)

$1 < x < y$.

N. N. LEBEDEV $G(x, y, \tau) = 0$. By subsequently setting $x = \cosh \alpha$ and $y = \cosh \alpha'$, we can write formula (25) in the form:

$\Gamma = \int f(\cosh \alpha') G(\cosh \alpha, \cosh \alpha', \tau) \sinh \alpha' \, d\alpha' =$

• j / ( c h a ' ) G ( c h a , c h a ' , T ) s h a ' d a ' =

$$I = I_1(\lambda, \Gamma) + I_2(\lambda, \Gamma). \tag{32}$$

To calculate the limit of the integral as $\lambda \to \infty$, we utilize formula (31) and the asymptotic representations given in (21). On the arc $\Gamma_2$,

$-\frac{\pi}{2} < \phi < \frac{\pi}{2}$; therefore,

(cha) Q i (cha')Q i (cha') I -2v(a4-a') —2v(a—a') 4 sh a s h a' I £—2v(a-}-a') j + cth a cth

Substituting $(33)$ into the first integral of $(31)$, we obtain:

$$ \int_{x}^{1} P_{\nu}(y) \, dy = \frac{1 - x^2}{2\nu + 1} \left[ P_{\nu-1}(x) - P_{\nu+1}(x) \right] $$

= 1 +

We have $-2va'$, from which it follows:

$$ -2va - 2v(a + a') / \cosh(T) $$

2va' f 0 ( 1 ) , 0 < a ' < 5 , dv = <! , j

V б <C a' < a, г \ 6 Г я J v г,

depends similarly

d v = 0 ( 7 , - x ) ,

1 f> e -2v (a+a')

dv = 0(T- 1 )

for all $0 < a'$. From this, we conclude that

$$\int X P_i(y) Q_1(y) dy =$$

LEBEDEV $1 + o(1)$

$0 < a' \le b$

1 + 0

• $\coth a' \sqrt{c} W_0 (T - 1/2)$, where $0 < a' < a$.

Thus, based on $V(V - \dots) + \coth(1/2)(r - 1/2) < a' < a$, it follows from (36) that

G ( c h a , c h a ' , T) =

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1 + 0

• cth a ' c h a ' С Н Г - 1 / 2 ) , 6 < a ' < a — б -

By partitioning the integration interval into the sub-intervals $(0, \delta)$, $(\delta, a - \delta)$, and $(a - \delta, a)$, and applying equation (38), we obtain:

$$I = J_x(x, T) = \int f(\cosh a') \sinh a' \, da' - f_0$$

• 0(1) | / ( c h a ' ) | s h a ' d a ' + 0(1) ) / ( c h a ' ) ' | s h a ' d a ' (tT~) ' ) i + 0 ( T - | / ( c h a ' ) | c h

x chS J,(x, T)= [f(y)dy+ 0 ( 1 ) j |/(y)|cfy +

\f(y)\dy+o(-^~) \f(y)\dy

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Taking $\gamma$ to be sufficiently small and then increasing $T$, we can make all integrals (with the exception of the first) arbitrarily small. Therefore, by an analogous process based on equations (30) and (34), and under the assumption that the second condition of (32) is satisfied, the theorem is proven. Thus, we have established the following result.

$\imJ(x, \Gamma) = \int \int f(y)dy$, (42)

This concludes the proof. In conclusion, it should be noted that the conditions imposed on the function $f(x)$ are sufficient; however, the theorem may hold under less restrictive assumptions. In particular, the theorem remains valid for functions satisfying the following conditions:

$f(x) - y = m'$

$f(x) \ln x \in L^a$, where $a > 1$ is a certain constant. The validity of this assertion regarding the function can be verified directly, after which the possibility of extending Theorem (3) to this class of functions becomes evident.

§ 3. EXAMPLES OF EXPANDING FUNCTIONS INTO INTEGRALS

On Squares of Legendre Functions: We present several interesting examples of expansions of this type, which can be derived from the general formula through an appropriate selection of the function.

$\frac{1}{x} \int_{0}^{\infty} \tau \tanh \pi \tau P_{-\frac{1}{2}+i\tau}(x) d\tau = \frac{1}{x}$

Here, the first of the conditions (2) is essential.

LEBEDEV $\frac{x+1}{2}$

$\int_{0}^{\infty} \tau \tanh \pi \tau P_{-\frac{1}{2}+i\tau}(x) \Gamma\left(\frac{1}{4} + \frac{i\tau}{2}\right) \Gamma\left(\frac{1}{4} - \frac{i\tau}{2}\right) d\tau = \frac{\sqrt{2}\pi}{\sqrt{x^2-1}}$

$(46)$ $\frac{1}{\sqrt{2x}}$

$a > -1$,

$\arcsin \sqrt{1 - a}$

$-1 < a < 1$,

— \ + Va + r t ~ T + ' «

$a > 1$,

$\int_{X} \Gamma(v-IT) [P^*] dx,$

1 Re

In these formulas, the function on the left side of the equality is related to the function $\phi(x)$ by the relation $f(x) = [\phi(x)]'$. The variable $x$ is defined such that the resulting equalities remain valid even at $x = 1$.

The values of the function $\Phi(m)$, defined by equality (4), are determined by replacing the product of spherical functions with an integral and utilizing the Mehler–Fock transform tables \cite{4, 5}.

References

Mehler, F. G. "Ueber eine Kugel- und Cylinder-Funktionen verwandte Funktion und ihre Anwendung in der Theorie der Elektricitäts-verteilung." Mathematische Annalen, Vol. 18. Fock, V. A. Doklady Akademii Nauk SSSR, 279–282. Lebedev, N. N. Special Functions and Their Applications. Moscow, 1963.

Tables of Lebedev, Mehler, and Generalized Mehler Transforms. Boeing Scientific Research Laboratories, Tables of Integral Transforms, Vol. II, 1954. Ioffe Physico-Technical Institute. Received July 14, 1966.