430 FUNCTIONS OF COMPLEX VARIABLE 10.2.2-3. Cauchy-type integral. Suppose that C is an arbitrary curve without cusps, not necessarily closed. Let an arbitrary function f(ξ), which is assumed to be finite and integrable, be given on this curve. The integral F (z)= 1 2πi C f(ξ) dξ ξ – z (10.2.2.13) is called a Cauchy-type integral. The Cauchy-type integral is a function analytic at any point z that does not lie on C. If the curve C divides the plane into several domains, then, in general, the Cauchy-type integral defines different analytic functions in these domains. One says that the function f(ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at a point ξ = ξ 0 of the contour C if there exists a constant M such that the inequality |f(ξ)–f(ξ 0 )| ≤ M|ξ – ξ 0 | μ (0 < μ ≤ 1)(10.2.2.14) holds for all points ξ C sufficiently close to ξ 0 .TheH ¨ older condition means that the increment of the function is an infinitesimal of order at least μ with respect to the increment of the argument. The principal value of the integral is defined as the limit lim r→0 C–c f(ξ) dξ ξ – ξ 0 = C f(ξ) dξ ξ – ξ 0 ,(10.2.2.15) where c is the segment of the curve C between the points of intersection of C with the circle |z – ξ 0 | = r. The singular integral in the sense of the Cauchy principal value is defined as the integral given by the formula C–c f(ξ) dξ ξ – ξ 0 = C f(ξ)–f(ξ 0 ) ξ – ξ 0 dξ + f(ξ 0 )ln b – ξ 0 a – ξ 0 + iπf(ξ 0 )+O(r), (10.2.2.16) where a and b are the endpoints of C and O(r) → 0 as r → 0. T HEOREM. If the function f(ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at a point ξ 0 which is a regular (nonsingular) point of the contour C and does not coincide with its endpoints, then the Cauchy-type integral exists at this point as a singular integral and its principal value can be expressed in terms of the usual integral by the formula F (ξ 0 )= 1 2πi C f(ξ) dξ ξ – ξ 0 = 1 2πi C f(ξ)–f(ξ 0 ) ξ – ξ 0 dξ + f(ξ 0 ) 2 + f(ξ 0 ) 2πi ln b – ξ 0 a – ξ 0 .(10.2.2.17) If the curve C is closed, then a = b and formula (10.2.2.17) becomes F (ξ 0 )= 1 2πi C f(ξ) dξ ξ – ξ 0 = 1 2πi C f(ξ)–f (ξ 0 ) ξ – ξ 0 dξ + f(ξ 0 ) 2 .(10.2.2.18) Suppose that the function f (ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at the point ξ = ξ 0 and the point z tends to ξ 0 so that the ratio of h = |z – ξ 0 | to d (dh is the shortest distance from z to the points of C) remains bounded. Then lim z→ξ 0 C f(ξ)–f (ξ 0 ) ξ – z dξ = C f(ξ)–f (ξ 0 ) ξ – ξ 0 dξ.(10.2.2.19) 10.2. MAIN APPLICATIONS 431 S OKHOTSKII’S THEOREM. Suppose that ξ 0 is a regular (nonsingular) point of the con- tour C and does not coincide with its endpoints, the function f(ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at this point, and z → ξ 0 so that the ratio h/d remains bounded. Then the Cauchy-type integral has limit values F + (ξ 0 ) and F – (ξ 0 ) to which this integral tends as z → ξ 0 from the left and, respectively, from the right of C ,and F + (ξ 0 )=F (ξ 0 )+ 1 2 f(ξ 0 ), F – (ξ 0 )=F (ξ 0 )– 1 2 f(ξ 0 ), (10.2.2.20) where F (ξ 0 ) is the singular integral (10.2.2.18). The Cauchy-type integral experiences a jump when passing through the integration contour C at the point ξ 0 : F + (ξ 0 )–F – (ξ 0 )=f(ξ 0 ). (10.2.2.21) The condition F – (ξ)=0 (10.2.2.22) at each point of C is necessary and sufficient for a Cauchy-type integral to be the Cauchy integral. T HEOREM. If a function f(ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at each point of a closed contour C , then, for its values to be the boundary values of a function analytic in the interior of C , it is necessary and sufficient that C ξ n f(ξ) dξ = 0 (n = 0, 1, 2, ). (10.2.2.23) T HEOREM. If a function f(ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at each point of a closed contour C , then, for the values of f(ξ) to be the boundary values of a function analytic in the interior of C , it is necessary and sufficient that 1 2πi C f(ξ) dξ ξ – z = 0 (10.2.2.24) for all points z lying in the exterior of C . THEOREM. If a function f(ξ) satisfies the H ¨ older condition with exponent μ ≤ 1 at each point of a closed contour C , then, for the values of f(ξ) to be the boundary values of a function analytic in the exterior of C , it is necessary and sufficient that 1 2πi C f(ξ) dξ ξ – z = f (∞)(10.2.2.25) for all points z lying in the interior of C . THEOREM. For the values of a function f(ξ) satisfying the H ¨ older condition with ex- ponent μ ≤ 1 to be the boundary values of a function analytic (a) in the interior of the disk |z| < 1 or (b) in the exterior of this disk, it is necessary and sufficient that the following respective conditions hold: for all z in the interior of C, 1 2πi C f(ξ) dξ ξ – z = f (0), (10.2.2.26) for all z in the exterior of C 1 2πi C f(ξ) dξ ξ – z = 0.(10.2.2.27) 432 FUNCTIONS OF COMPLEX VARIABLE Example (the first main problem of elasticity). Let D be the unit disk. Find the elastic equilibrium for given external stresses F n = X n + iY n on the unit circle C,whereX n and Y n are the components of a surface force vector. The problem is to find functions ϕ and ψ satisfying the boundary condition ϕ(ξ)+ξ ϕ ξ (ξ)+ψ(ξ)=f(ξ), where f(ξ)=i ξ ξ 0 F n ds is a function given on C. To be definite, we set ψ(0)=Imϕ ξ (0)=0. By formula (10.2.2.26), the relation 1 2πi C ψ(ξ) dξ ξ – z = 1 2πi C f(ξ) dξ ξ – z – 1 2πi C ϕ(ξ) dξ ξ – z – 1 2πi C ξϕ ξ (ξ) ξ – z dξ = 0 holds for all |z| < 1. Since the function ϕ(z) is analytic in the disk |z| < 1, we can use the Cauchy formula and rewrite this relation as ϕ(z)+ 1 2πi C ξϕ ξ (ξ) ξ – z dξ = 1 2πi C f(ξ) dξ ξ – z . Thus we obtain an equation for the function ϕ(z). Omitting the details, we write out the definitive result: ϕ(z)= 1 2πi C f(ξ) dξ ξ – z – z 4πi C f(ξ) dξ ξ 2 . To find the function ψ(z), we pass from the boundary condition ϕ(ξ)+ξ ϕ ξ (ξ)+ψ(ξ)=f (ξ)tothecomplex conjugate condition and solve it for ψ(z). Thus we obtain ψ(ξ)= f(ξ)–ϕ(ξ)–ξϕ ξ (ξ). We calculate the Cauchy-type integral of the expressions in both sides, which is reduced to the Cauchy integral in either case, and obtain ψ(z)= 1 2πi C f(ξ) dξ ξ – z + 1 4πiz C f(ξ) dξ ξ 2 – ϕ ξ (ξ) z . 10.2.2-4. Hilbert–Privalov boundary value problem. Privalov boundary value problem. Given two complex functions a(ξ) ≠ 0 and b(ξ) satisfying the H ¨ older condition with exponent μ ≤ 1 on a closed curve C, find a function f – (z) analytic in the exterior of C including the point at infinity z = ∞and a function f + (z) analytic in the interior of C such that the boundary values f – (ξ)andf + (ξ) of these functions on C exist and satisfy the relation f – (ξ)=a(ξ)f + (ξ)+b(ξ). (10.2.2.28) If b(ξ)=0, i.e., if the boundary relation has the form f – (ξ)=a(ξ)f + (ξ), (10.2.2.29) then the Privalov boundary value problem is called the Hilbert boundary value problem. The index (winding number) of a function a(ξ)isdefined to be the integer equal to the net increment of its argument along the closed curve C,dividedby2π: 1 2π Δ C arg a(ξ)= 1 2πi C d ln a(ξ). (10.2.2.30) G AKHOV’S FIRST THEOREM. The Hilbert problem f – (ξ)=a(ξ)f + (ξ) has a family of solutions depending on n + 1 arbitrary constants if the index n of the boundary function a(ξ) is not positive. If the index n is positive, then the problem does not have solutions analytic in the corresponding domains. REFERENCES FOR CHAPTER 10 433 The solutions of the Hilbert problem can be written as f – (ξ)= a 0 + a 1 z + ···+ a n z n exp[–F – 1 (z)], f + (ξ)=(a 0 z n + a 1 z n–1 + ···+ a n )exp[–F + 1 (z)], (10.2.2.31) where F 1 (z)= 1 2πi C ln[ξ n a(ξ)] ξ – z dξ. The constants a 0 , , a n in formula (10.2.2.31) are arbitrary, and a 0 is determined by the choice of the value f – (∞). G AKHOV’S SECOND THEOREM. The Privalov problem f – (ξ)=a(ξ)f + (ξ)+b(ξ) has a family of solutions depending on n + 1 arbitrary constants if the index n of the boundary function a(ξ) is not positive. If the index n of the function a(ξ) is positive, then the problem is solvable only if the function b(ξ) satisfies the condition C b(ξ)exp[–F – 1 (ξ)] ξ k+1 dξ = 0 (k = 1, 2, , n). (10.2.2.32) The solutions of the Privalov problem can be written as f – (ξ)= a 0 + a 1 z + + a n z n + F – 2 (z) exp[–F – 1 (z)], f + (ξ)= a 0 z n + a 1 z n–1 + + a n + z n F + 2 (z) exp[–F + 1 (z)], (10.2.2.33) where a 0 , , a n are arbitrary constants and F 2 (z) is determined by the formula F 2 (z)=– 1 2πi C b(ξ)exp[–F 1 (ξ)] ξ – z dξ. References for Chapter 10 Ablowitz, M. J. and Fokas, A. S., Complex Variables: Introduction and Applications (Cambridge Texts in Applied Mathematics), 2nd Edition, Cambridge University Press, Cambridge, 2003. Berenstein, C. A. and Roger Gay, R., Complex Variables: An Introduction (Graduate Texts in Mathematics), Springer, New York, 1997. Bieberbach, L., Conformal Mapping, American Mathematical Society, Providence, Rhode Island, 2000. Bronshtein, I. N., Semendyayev, K. A., Musiol, G., and M ¨ uhlig, H., Handbook of Mathematics, 4th Edition, Springer, New York, 2004. Brown, J. W. and Churchill, R. V., Complex Variables and Applications, 7th Edition, McGraw-Hill, New York, 2003. Caratheodory, C., Conformal Representation, Dover Publications, New York, 1998. Carrier, G. F., Krock, M., and Pearson, C. E., Functions of a Complex Variable: Theory and Technique (Classics in Applied Mathematics), Society for Industrial & Applied Mathematics, University City Science Center, Philadelphia, 2005. Cartan, H., Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover Publications, New York, 1995. Conway,J.B.,Functions of One Complex Variable I (Graduate Texts in Mathematics), 2nd Edition, Springer, New York, 1995. 434 FUNCTIONS OF COMPLEX VARIABLE Conway, J. B., Functions of One Complex Variable II (Graduate Texts in Mathematics), 2nd Edition, Springer, New York, 1996. Dettman,J.W.,Applied Complex Variables (Mathematics Series), Dover Publications, New York, 1984. England, A. H., Complex Variable Methods in Elasticity, Dover Edition, Dover Publications, New York, 2003. Fisher, S. D., Complex Variables (Dover Books on Mathematics), 2nd Edition, Dover Publications, New York, 1999. Flanigan, F. J., Complex Variables, Dover Ed. Edition, Dover Publications, New York, 1983. Greene, R. E. and Krantz, S. G., Function Theory of OneComplex Variable (Graduate StudiesinMathematics), Vol. 40, 2nd Edition, American Mathematical Society, Providence, Rhode Island, 2002. Ivanov,V.I.andTrubetskov,M.K.,Handbook of Conformal Mapping with Computer-Aided Visualization, CRC Press, Boca Raton, 1995. Korn, G. A and Korn, T. M., Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Dover Edition, Dover Publications, New York, 2000. Krantz, S. G., Handbook of Complex Variables,Birkh ¨ auser, Boston, 1999. Lang, S., Complex Analysis (Graduate Texts in Mathematics), 4th Edition, Springer, New York, 2003. Lavrentiev,M.A.andShabat,V.B.,Methods of the Theory of Functions of a Complex Variable, 5th Edition [in Russian], Nauka Publishers, Moscow, 1987. LePage, W. R., Complex Variables and the Laplace Transform for Engineers, Dover Publications, New York, 1980. Markushevich, A. I. and Silverman, R. A. (Editor), Theory of Functions of a Complex Variable, 2nd Rev. Edition, American Mathematical Society, Providence, Rhode Island, 2005. Narasimhan, R. and Nievergelt, Y., Complex Analysis in One Variable, 2nd Edition,Birkh ¨ auser, Boston, Basel, Stuttgard, 2000. Needham, T., Visual Complex Analysis, Rep. Edition, Oxford University Press, Oxford, 1999. Nehari, Z., Conformal Mapping, Dover Publications, New York, 1982. Paliouras, J. D. and Meadows, D. S., Complex Variables for Scientists and Engineers, Facsimile Edition, Macmillan Coll. Div., New York, 1990. Pierpont, J., Functions of a Complex Variable (Phoenix Edition), Dover Publications, New York, 2005. Schinzinger, R. and Laura, P. A. A., Conformal Mapping: Methods and Applications, Dover Publications, New York, 2003. Silverman, R. A., Introductory Complex Analysis, Dover Publications, New York, 1984. Spiegel, M. R., Schaum’s Outline of Complex Variables, McGraw-Hill, New York, 1968. Sveshnikov, A. G. and Tikhonov, A. N., The Theory of Functions of a Complex Variable, Mir Publishers, Moscow, 1982. Wunsch, D. A., Complex Variables with Applications, 2nd Edition, Addison Wesley, Boston, 1993. Chapter 11 Integral Transforms 11.1. General Form of Integral Transforms. Some Formulas 11.1.1. Integral Transforms and Inversion Formulas Normally an integral transform has the form f(λ)= b a ϕ(x, λ)f(x) dx.(11.1.1.1) The function f(λ) is called the transform of the function f(x)andϕ(x, λ) is called the kernel of the integral transform. The function f (x) is called the inverse transform of f(λ). The limits of integration a and b are real numbers (usually, a = 0, b = ∞ or a =–∞, b = ∞). For brevity, we rewrite formula (11.1.1.1) as follows: f(u)=L{f(x)}. General properties of integral transforms (linearity): L{kf(x)} = kL{f (x)} , L{f(x) g(x)} = L{f (x)} L{g(x)}. Here, k is an arbitrary constant; it is assumed that integral transforms of the functions f(x) and g(x)exist. In Subsections 11.2–11.6, the most popular (Laplace, Mellin, Fourier, etc.) integral transforms are described. These subsections also describe the corresponding inversion formulas, which normally have the form f(x)= C ψ(x, λ) f(λ) dλ (11.1.1.2) and make it possible to recover f (x)if f(λ) is given. The integration path C can lie either on the real axis or in the complex plane. In many cases, to evaluatethe integrals in the inversion formula (11.1.1.2)—in particular, to find the inverse Laplace, Mellin, and Fourier transforms —methods of the theory of functions of a complex variable can be applied, including the residue theorem and the Jordan lemma, which are briefly outlined below in Subsection 11.1.2. 11.1.2. Residues. Jordan Lemma 11.1.2-1. Residues. Calculation formulas. The residue of a function f(z) holomorphic in a deleted neighborhood of a point z = a (thus, a is an isolated singularity of f )ofthecomplexplanez is the number res z=a f(z)= 1 2πi C ε f(z) dz, i 2 =–1, where C ε is a circle of sufficiently small radius ε described by the equation |z – a| = ε. 435 436 INTEGRAL TRANSFORMS If the point z = a is a pole of order n* of the function f(z), then we have res z=a f(z)= 1 (n – 1)! lim z→a d n–1 dx n–1 (z – a) n f(z) . For a simple pole, which corresponds to n = 1, this implies res z=a f(z) = lim z→a (z – a)f(z) . If f(z)= ϕ(z) ψ(z) ,whereϕ(a) ≠ 0 and ψ(z) has a simple zero at the point z = a, i.e., ψ(a)=0 and ψ z (a) ≠ 0,then res z=a f(z)= ϕ(a) ψ z (a) . 11.1.2-2. Jordan lemma. If a function f(z) is continuous in the domain |z| ≥ R 0 ,Imz ≥ α,whereα is a chosen real number, and if lim z→∞ f(z)=0,then lim R→∞ C R e iλz f(z) dz = 0 for any λ > 0,whereC R is the arc of the circle |z| = R that lies in this domain. For more details about residues and the Jordan lemma, see Paragraphs 10.1.2-7 and 10.1.2-8. 11.2. Laplace Transform 11.2.1. Laplace Transform and the Inverse Laplace Transform 11.2.1-1. Laplace transform. The Laplace transform of an arbitrary (complex-valued) function f(x) of a real variable x (x ≥ 0)isdefined by f(p)= ∞ 0 e –px f(x) dx,(11.2.1.1) where p = s + iσ is a complex variable. The Laplace transform exists for any continuous or piecewise-continuous function satisfying the condition |f(x)| < Me σ 0 x with some M > 0 and σ 0 ≥ 0. In the following, σ 0 often means the greatest lower bound of the possible values of σ 0 in this estimate; this value is called the growth exponent of the function f (x). For any f(x), the transform f(p)isdefined in the half-plane Re p > σ 0 and is analytic there. For brevity, we shall write formula (11.2.1.1) as follows: f(p)=L f(x) ,or f(p)=L f(x), p . * In a neighborhood of this point we have f(z) ≈ const (z – a) –n . . 2002. Ivanov,V.I.andTrubetskov,M.K. ,Handbook of Conformal Mapping with Computer-Aided Visualization, CRC Press, Boca Raton, 1995. Korn, G. A and Korn, T. M., Mathematical Handbook for Scientists and Engineers: Definitions,. 1993. Chapter 11 Integral Transforms 11.1. General Form of Integral Transforms. Some Formulas 11.1.1. Integral Transforms and Inversion Formulas Normally an integral transform has the form f(λ)= b a ϕ(x,. the transform of the function f(x )and (x, λ) is called the kernel of the integral transform. The function f (x) is called the inverse transform of f(λ). The limits of integration a and b are