UDC 517.53:512.9

MATHEMATICS

G. N. POLOZHII

ON INVERSION FORMULAS FOR THE FUNDAMENTAL INTEGRAL REPRESENTATION OF \(p\)-ANALYTIC FUNCTIONS OF \(z=x+iy\) WITH CHARACTERISTIC \(p=x^k\)

(Presented by Academician I. N. Vekua, 12 I 1967)

Let \(G\) be a domain in the right half-plane \(z=x+iy\), and let \(a\) be a point on its boundary \(S\). The fundamental integral representation of \(x^k\)-analytic functions of \(z=x+iy\) \((x>0,\ k=\mathrm{const}>0)\) has the form \((^{1-3})\)

\[ \begin{aligned} \widetilde f(z) &=\widetilde u(x,y)+i\widetilde v(x,y) \\ &=\operatorname{Re}\int_a^z f(\zeta) \left(\frac{z+\bar z}{2}\right)^{1-k} (z-\zeta)^{k/2-1}(\bar z+\zeta)^{k/2-1}\,d\zeta \\ &\quad + i\,\operatorname{Im}\int_a^z f(\zeta) \left(\zeta-\frac{z-\bar z}{2}\right) (z-\zeta)^{k/2-1}(\bar z+\zeta)^{k/2-1}\,d\zeta , \end{aligned} \tag{1} \]

where the integration from \(a\) to \(z\) is carried out along any contour \(\Gamma\) lying in \(G\); \(\arg (z-\zeta)(\bar z+\zeta)\) is chosen in one way or another; \(f(z)=u(x,y)+iv(x,y)\) and \(\widetilde f(z)=\widetilde u(x,y)+i\widetilde v(x,y)\) are analytic and, respectively, \(x^k\)-analytic functions in the domain \(G\). In this case three cases are essentially distinguished: a) \(S\) contains a segment \(L\) of the imaginary axis and \(v|_L=0\), while \(a\) is an arbitrary (unfixed) point on \(L\); b) \(a=\infty\) and, as \(z\to\infty\), \(f(z)=O\bigl(|z|^{-k-\varepsilon}\bigr)\) \((\varepsilon=\mathrm{const}>0)\); c) \(a\ne\infty\) and, as \(z\to a\), \(f(z)=O\bigl(|z-a|^{-1+\varepsilon}\bigr)\).

Introducing the “adjoint” \(p\)-analytic function with characteristic \(p=x^k\),

\[ f^*(z)=u^*(x,y)+iv^*(x,y)=\widetilde u(x,y)+\frac{1}{x^k}\,\widetilde v(x,y), \]

equality (1) can be written in the equivalent form

\[ \begin{aligned} f^*(z) &=u^*(x,y)+iv^*(x,y) \\ &=\frac{1}{2x^k}\int_a^z f(\zeta)(z-\zeta)^{k/2-1}(\bar z+\zeta)^{k/2}\,d\zeta +\overline{f(\zeta)}(\bar z-\bar\zeta)^{k/2}(z+\bar\zeta)^{k/2-1}\,d\bar\zeta . \end{aligned} \tag{2} \]

From the conditions of \(p\)-analyticity it follows that \((^{4,5})\)

\[ \frac{\partial}{\partial \bar z}f^* =-\frac{k}{2(z+\bar z)}\,(f^*-\bar f^*), \tag{3} \]

i.e. \(f^*(z)\) is a generalized analytic function \((^6)\) or a generalized analytic function of the fourth class \((^5)\).

In cases b) and c) we introduce some additional assumptions. Instead of case b) we shall speak of case b′), assuming that in \(G\) there exists a rectilinear horizontal segment \(C\) issuing from the point \(a=\infty\), and that on this segment, as \(z\to\infty\), \(f(z),\,df(z)/dz=O(x^{-2-\varepsilon})\). From case c) we single out two cases c′) and c″), assuming that in \(G\) there exists a rectilinear horizontal segment \(C\) issuing from the point \(a\) to the right or, respectively, to the left, and that \(f(z),\,df(z)/dz=O(1)\) as \(z\to a\).

It is easy to establish that

\[ \widetilde F(z)=\widetilde U(x,y)+i\widetilde V(x,y)= \]

\[ = x^{k-1}\widetilde u(x,y)+i\int_a^z x^{1-k}\frac{\partial \widetilde v(x,y)}{\partial x}\,dx+ \left(x^{1-k}\frac{\partial \widetilde v(x,y)}{\partial y}+(k-1)\widetilde u(x,y)\right)dy \tag{4} \]

will be an \(x^{k'}\)-analytic function in \(G\), where \(k'=2-k\).

If \(0<k<2\) and

\[ F^*(z)=U^*(x,y)+iV^*(x,y)=\widehat U(x,y)+i\frac{1}{x^k}V(x,y) \]

is the adjoint \(x^{k'}\)-analytic function, then the integral

\[ I(z)=\int_a^z F^*(t)(t-z)^{k'/2-1}(\bar t+z)^{k'/2}\,dt+ \overline{F^*(t)}(t-z)^{k'/2}(\bar t+z)^{k'/2-1}\,d\bar t \tag{5} \]

does not depend on the contour of integration \(\Gamma\) connecting the points \(a\) and \(z\), and is a function of \(z\) analytic in \(G\).

The first part of the assertion follows from the identity (5)

\[ \frac{\partial}{dt}\left(F^*(t)(t-z)^{k'/2-1}(\bar t+z)^{k'/2}\right) = \frac{\partial}{d\bar t}\left(\overline{F^*(t)}(t-z)^{k'/2}(\bar t+z)^{k'/2-1}\right), \]

the second part of the assertion—from the equality

\[ I(z)\left(1-e^{\mp i2\pi(k'/2-1)}\right) = \int_\gamma F^*(t)(t-z)^{k'/2-1}(\bar t+z)^{k'/2}\,dt+ \]

\[ +\overline{F^*(t)}(t-z)^{k'/2}(\bar t+z)^{k'/2-1}\,d\bar t, \]

where \(\gamma\) is the contour going from the point \(a\) along the left (or right) edge of the cut along \(\Gamma\), encircling the point \(z\) in the negative (or positive) direction and returning to the point \(a\) along the right (or left) edge of the cut.

A. Consider cases a) and b′). Taking \(\arg (z-\zeta)(\bar z+\zeta)=0\) on the upper (left) edge of the cut along \(C\) and understanding by the values of the functions their values on the indicated edge of the cut along \(C\) or \(\Gamma\), introduce the function analytic in \(G\)

\[ f_1(z)=u_1(x,y)+iv_1(x,y)=\mu\,\frac{1}{2e^{i\pi(k'/2-1)}}\frac{d}{dz}I(z), \tag{6} \]

where \(\mu=2\Gamma^{-1}(-k/2+1)\Gamma^{-1}(k/2)\).

On the segment \(C\), equalities (6) and (1) take the form

\[ f_1(z)=u_1(x,y)+iv_1(x,y)= \]

\[ =\mu\frac{d}{dx}\int_{x_0}^{x}\left[\widetilde u(\xi,y)\xi^k+i\frac{x}{\xi^{2-k}}V(\xi,y)\right] \frac{d\xi}{(x^2-\xi^2)^{k/2}}, \tag{7} \]

\[ \widetilde f(z)=\widetilde u(x,y)+i\widetilde v(x,y) = \int_{x_0}^{x}\left[x^{1-k}u(\xi,y)+i\xi v(\xi,y)\right](x^2-\xi^2)^{k/2-1}\,d\xi, \tag{8} \]

where \(x_0=\operatorname{Re} a\).

From (8) and (4) it follows that

\[ \left.\widetilde v(x,y)\right|_{z=a}=0,\qquad \left.\widetilde V(x,y)\right|_{z=a}=0. \tag{9} \]

The inversion formula (8) has the form \((^{5,7})\)

\[ u(x,y)+ixv(x,y)= \mu\frac{d}{dx}\int_{x_0}^{x} \left[i\widetilde u(\xi,y)\xi^{k-1}+\widetilde v(\xi,y)\right] \frac{\xi\,d\xi}{(x^2-\xi^2)^{k/2}}. \tag{10} \]

Comparing (7), (10) and taking (9) into account, we find on \(C\)

\[ f_1(z)=f(z)=\mu {d\over dx}\int_{x_0}^{x}\widetilde u(\xi,y)\, {\xi^k\,d\xi\over (x^2-\xi^2)^{k/2}} +i\mu x^{1-k}\int_{x_0/x}^{1}{\partial \widetilde v(x\beta,y)\over \partial x\beta}\, {d\beta\over (1-\beta^2)^{k/2}}. \]

Consequently, \(f_1(z)\equiv f(z)\) in the domain \(G\), and the desired inversion formula for the basic integral representation (1) is written in the form

\[ f(z)=u(x,y)+iv(x,y)= \tag{11} \]

\[ =\mu {1\over 2e^{i\pi(k'/2-1)}}{d\over dz} \int_a^z F^*(t)(t-z)^{k'/2-1}(\bar t+z)^{k'/2}\,dt +\bar F^*(t)(t-z)^{k'/2}(\bar t+z)^{k'/2-1}\,d\bar t \]

or

\[ f(z)=u(x,y)+iv(x,y)= \mu {1\over 2e^{i\pi(k'/2-1)}}{d\over dz} \int_a^z \widetilde U(\xi,\eta)\,d\widetilde Z +i\widetilde V(\xi,\eta)\,dZ, \tag{12} \]

where \(Z\) and \(\widetilde Z\) are the conjugate variables \((^4,^5)\) of \(t=\xi+i\eta\) with characteristic \(p=\xi^{k'}\),

\[ \widetilde Z={2\over k'}(t-z)^{k'/2}(\bar t+z)^{k'/2},\quad dZ=2^{k'+1}\left(\sqrt{{t-z\over t+z}}+ \sqrt{\left({\bar t+z\over t-z}\right)^{-k'}}\right) d\ln\sqrt{{t-z\over \bar t+z}}. \tag{12'} \]

B. Let us consider cases б′) and в″). Setting \(\arg (z-\xi)(\bar z+\xi)=-\pi\) on the upper (right-hand) edge of the cut along \(C\) and understanding by the values of the functions their values on the indicated edge of the cut along \(C\) or \(\Gamma\), introduce the function analytic in \(G\)

\[ f_2(z)=u_2(x,y)+iv_2(x,y)=\mu {1\over 2}{d\over dz}I(z). \tag{13} \]

From (13) and (1) we obtain on \(C\)

\[ f_2(z)=u_2(x,y)+iv_2(x,y)= \mu {d\over dx}\int_{x_0}^{x} \left[\widetilde u(\xi,y)\xi^k+i{x\over \xi^{2-k}}\widetilde V(\xi,y)\right] {d\xi\over (\xi^2-x^2)^{k/2}}, \tag{14} \]

\[ \widetilde f(z)=\widetilde u(x,y)+i\widetilde v(x,y)= \int_{x_0/x}^{1}[u_3(x\beta,y)+ix^k\beta v_3(x\beta,y)](\beta^2-1)^{k/2-1}\,d\beta, \tag{15} \]

where \(f_3(z)=u_3(x,y)+iv_3(x,y)=e^{-i\pi(k/2-1)}f(z)\). From (15) it follows that in case б′), when approaching infinity,

\[ \widetilde u(x,y),\quad \partial\widetilde u(x,y)/\partial x,\quad \partial\widetilde u(x,y)/\partial y=O(x^{-2-\varepsilon}),\quad \widetilde v(x,y)=O(x^{k-2-\varepsilon}). \]

Therefore in case б′), as in case в″), the equalities (9) hold. Taking this and the inversion formula for the basic integral representation (1) on \(C\) \((^5,^7)\) into account,

\[ u_3(x,y)+ixv_3(x,y)= \mu {d\over dx}\int_{x_0}^{x} [\widetilde u(\xi,y)\xi^{k-1}+i\widetilde v(\xi,y)] {\xi\,d\xi\over (\xi^2-x^2)^{k/2}}, \]

we come to the conclusion that on \(C\)

\[ f_2(z)=f_3(z)= \mu {d\over dx}\int_{x_0}^{x}\widetilde u(\xi,y)\, {\xi^k\,d\xi\over (\xi^2-x^2)^{k/2}} +i\mu x^{1-k}\int_{x_0/x}^{1} {\partial \widetilde v(x\beta,y)\over \partial x\beta} {d\beta\over (\beta^2-1)^{k/2}}. \]

Thus, the desired inversion formula for the basic integral representation (1) is obtained in the following form:

\[ f(z)=u(x,y)+iv(x,y)= \]

\[ = \mu \frac{1}{2e^{i\pi k'/2}}\,\frac{d}{dz} \int_a^z F^*(t)(t-z)^{k'/2-1}(\bar t+z)^{k'/2}\,dt \]
\[ +\bar F^*(t)(t-z)^{k'/2}(\bar t+z)^{k'/2-1}\,d\bar t \tag{16} \]

or

\[ f(z)=u(x,y)+iv(x,y)= \mu \frac{1}{2e^{i\pi k'/2}}\,\frac{d}{dz} \int_a^z \widetilde U(\xi,\eta)\,d\widetilde Z+i\widetilde V(\xi,\eta)\,dZ, \tag{17} \]

where \(Z\) and \(\widetilde Z\) are conjugate variables defined by the equalities \((12')\), under the agreement made above on the choice of \(\arg (z-\zeta)(\bar z+\zeta)\).

C). Let now \(k=2m+k_1\), where \(m\) is a positive integer, \(0<k_1<2\). From (2) it follows that the function

\[ f_1^*(z)=u_1^*(x,y)+iv_1^*(x,y)= \frac{1}{\alpha x^{k_1}}\, \frac{d^{2m}}{dz^m\,d\bar z^m}\bigl(x^k f^*(z)\bigr), \]
\[ \alpha=\left(\frac{k}{2}-1\right)\frac{k}{2} \left(\frac{k}{2}-2\right)\left(\frac{k}{2}-1\right)\cdots \frac{1}{2}\,\frac{3}{2} \tag{18} \]

will be adjoint \(x^{k_1}\)-analytic in \(G\). Introducing the \(x^{k_1}\)-analytic function
\[ f_1(z)=\widetilde u_1(x,y)+i\widetilde v_1(x,y) =u_1^*(x,y)+ix^{k_1}v_1^*(x,y), \]
we arrive at the conclusion that the inversion formula of the principal integral representation (1) for \(k=2m+k_1\) is given by equality (11) or (12) in cases a) and b′), and by equality (16) or (17) in cases b) and c′), provided that in these equalities \(k\) is replaced by \(k_1\), and \(\widetilde u(x,y)\) and \(\widetilde v(x,y)\) are replaced by \(\widetilde u_1(x,y)\) and \(\widetilde v_1(x,y)\).

Kyiv State University
named after T. G. Shevchenko

Received
4 I 1967

CITED LITERATURE

¹ G. M. Polozhii, Scientific Yearbook of Kyiv University for 1957, Kyiv, 1958, p. 340.
² G. M. Polozhii, Bulletin of Kyiv University, No. 2, series astr., matem., mekh., issue 1, 1959, p. 19.
³ G. N. Polozhii, Ukrainian Mathematical Journal, 16, issue 2 (1964).
⁴ G. N. Polozhii, DAN, 58, No. 7 (1947).
⁵ G. N. Polozhii, Generalization of the Theory of Analytic Functions of a Complex Variable. \(p\)-Analytic and \((p,q)\)-Analytic Functions and Some of Their Applications, Kyiv, 1965.
⁶ I. N. Vekua, Generalized Analytic Functions, Moscow, 1959.
⁷ G. N. Polozhii, Ukrainian Mathematical Journal, 16, No. 5 (1964).