Handbook of mathematics for engineers and scienteists part 4 potx

CONTENTS xxi T5. Ordinary Differential Equations 1207 T5.1. First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 T5.2. Second-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1212 T5.2.1. Equations Involving Power Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1213 T5.2.2. Equations Involving Exponential and Other Functions . . . . . . . . . . . . . . . . . . . 1220 T5.2.3. Equations Involving Arbitrary Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1222 T5.3. Second-Order Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223 T5.3.1. Equations of the Form y  xx = f (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223 T5.3.2. Equations of the Form f(x, y)y  xx = g(x, y, y  x ) . . . . . . . . . . . . . . . . . . . . . . . . . 1225 References for Chapter T5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228 T6. Systems of Ordinary Differential Equations 1229 T6.1. Linear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 T6.1.1. Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1229 T6.1.2. Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1232 T6.2. Linear Systems of Three and More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1237 T6.3. Nonlinear Systems of Two Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239 T6.3.1. Systems of First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1239 T6.3.2. Systems of Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1240 T6.4. Nonlinear Systems of Three or More Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244 References for Chapter T6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246 T7. First-Order Partial Differential Equations 1247 T7.1. Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1247 T7.1.1. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = 0 . . . . . . . . . . . . . . . . . . . . . . 1247 T7.1.2. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y) . . . . . . . . . . . . . . . . . 1248 T7.1.3. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y)w + r(x, y) . . . . . . . . 1250 T7.2. Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1252 T7.2.1. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y, w) . . . . . . . . . . . . . . . 1252 T7.2.2. Equations of the Form ∂w ∂x + f(x, y, w) ∂w ∂y = 0 . . . . . . . . . . . . . . . . . . . . . . . . . 1254 T7.2.3. Equations of the Form ∂w ∂x + f(x, y, w) ∂w ∂y = g(x, y, w) . . . . . . . . . . . . . . . . . . 1256 T7.3. Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258 T7.3.1. Equations Quadratic in One Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1258 T7.3.2. Equations Quadratic in Two Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1259 T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives . . . . . . . . . . . . . . . . . . . 1261 References for Chapter T7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265 T8. Linear Equations and Problems of Mathematical Physics 1267 T8.1. Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267 T8.1.1. Heat Equation ∂w ∂t = a ∂ 2 w ∂x 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1267 T8.1.2. Nonhomogeneous Heat Equation ∂w ∂t = a ∂ 2 w ∂x 2 + Φ(x, t) . . . . . . . . . . . . . . . . . . 1268 T8.1.3. Equation of the Form ∂w ∂t = a ∂ 2 w ∂x 2 + b ∂w ∂x + cw + Φ(x, t) . . . . . . . . . . . . . . . . . 1270 T8.1.4. Heat Equation with Axial Symmetry ∂w ∂t = a  ∂ 2 w ∂r 2 + 1 r ∂w ∂r  . . . . . . . . . . . . . . . 1270 T8.1.5. Equation of the Form ∂w ∂t = a  ∂ 2 w ∂r 2 + 1 r ∂w ∂r  + Φ(r, t) . . . . . . . . . . . . . . . . . . . 1271 T8.1.6. Heat Equation with Central Symmetry ∂w ∂t = a  ∂ 2 w ∂r 2 + 2 r ∂w ∂r  . . . . . . . . . . . . . 1272 T8.1.7. Equation of the Form ∂w ∂t = a  ∂ 2 w ∂r 2 + 2 r ∂w ∂r  + Φ(r, t) . . . . . . . . . . . . . . . . . . . 1273 T8.1.8. Equation of the Form ∂w ∂t = ∂ 2 w ∂x 2 + 1–2β x ∂w ∂x . . . . . . . . . . . . . . . . . . . . . . . . . . . 1274 xxii CONTENTS T8.1.9. Equations of the Diffusion (Thermal) Boundary Layer . . . . . . . . . . . . . . . . . . . 1276 T8.1.10. Schr ¨ odinger Equation i ∂w ∂t =–  2 2m ∂ 2 w ∂x 2 + U(x)w . . . . . . . . . . . . . . . . . . . . . 1276 T8.2. Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278 T8.2.1. Wave Equation ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1278 T8.2.2. Equation of the Form ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 + Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1279 T8.2.3. Klein–Gordon Equation ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 – bw . . . . . . . . . . . . . . . . . . . . . . . . . . . 1280 T8.2.4. Equation of the Form ∂ 2 w ∂t 2 = a 2 ∂ 2 w ∂x 2 – bw + Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . 1281 T8.2.5. Equation of the Form ∂ 2 w ∂t 2 = a 2  ∂ 2 w ∂r 2 + 1 r ∂w ∂r  + Φ(r, t) . . . . . . . . . . . . . . . . . . 1282 T8.2.6. Equation of the Form ∂ 2 w ∂t 2 = a 2  ∂ 2 w ∂r 2 + 2 r ∂w ∂r  + Φ(r, t) . . . . . . . . . . . . . . . . . . 1283 T8.2.7. Equations of the Form ∂ 2 w ∂t 2 + k ∂w ∂t = a 2 ∂ 2 w ∂x 2 + b ∂w ∂x + cw + Φ(x, t) . . . . . . . . . 1284 T8.3. Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284 T8.3.1. Laplace Equation Δw = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284 T8.3.2. Poisson Equation Δw + Φ(x) = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1287 T8.3.3. Helmholtz Equation Δw + λw =–Φ(x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1289 T8.4. Fourth-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294 T8.4.1. Equation of the Form ∂ 2 w ∂t 2 + a 2 ∂ 4 w ∂x 4 = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294 T8.4.2. Equation of the Form ∂ 2 w ∂t 2 + a 2 ∂ 4 w ∂x 4 = Φ(x, t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1295 T8.4.3. Biharmonic Equation ΔΔw = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297 T8.4.4. Nonhomogeneous Biharmonic Equation ΔΔw = Φ(x, y) . . . . . . . . . . . . . . . . 1298 References for Chapter T8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299 T9. Nonlinear Mathematical Physics Equations 1301 T9.1. Parabolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1301 T9.1.1. Nonlinear Heat Equations of the Form ∂w ∂t = ∂ 2 w ∂x 2 + f(w) . . . . . . . . . . . . . . . . 1301 T9.1.2. Equations of the Form ∂w ∂t = ∂ ∂x  f(w) ∂w ∂x  + g(w) . . . . . . . . . . . . . . . . . . . . . . 1303 T9.1.3. Burgers Equation and Nonlinear Heat Equation in Radial Symmetric Cases . . 1307 T9.1.4. Nonlinear Schr ¨ odinger Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1309 T9.2. Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1312 T9.2.1. Nonlinear Wave Equations of the Form ∂ 2 w ∂t 2 = a ∂ 2 w ∂x 2 + f(w) . . . . . . . . . . . . . . 1312 T9.2.2. Other Nonlinear Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316 T9.3. Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318 T9.3.1. Nonlinear Heat Equations of the Form ∂ 2 w ∂x 2 + ∂ 2 w ∂y 2 = f(w) . . . . . . . . . . . . . . . . 1318 T9.3.2. Equations of the Form ∂ ∂x  f(x) ∂w ∂x  + ∂ ∂y  g(y) ∂w ∂y  = f(w) . . . . . . . . . . . . . . 1321 T9.3.3. Equations of the Form ∂ ∂x  f(w) ∂w ∂x  + ∂ ∂y  g(w) ∂w ∂y  = h(w) . . . . . . . . . . . . . . 1322 T9.4. Other Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324 T9.4.1. Equations of Transonic Gas Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324 T9.4.2. Monge–Amp ` ere Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1326 T9.5. Higher-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 T9.5.1. Third-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327 T9.5.2. Fourth-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1332 References for Chapter T9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1335 T10. Systems of Partial Differential Equations 1337 T10.1. Nonlinear Systems of Two First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 T10.2. Linear Systems of Two Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341 CONTENTS xxiii T10.3. Nonlinear Systems of Two Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1343 T10.3.1. Systems of the Form ∂u ∂t = a ∂ 2 u ∂x 2 + F (u, w), ∂w ∂t = b ∂ 2 w ∂x 2 + G(u, w) . . . . . . 1343 T10.3.2. Systems of the Form ∂u ∂t = a x n ∂ ∂x  x n ∂u ∂x  + F(u, w), ∂w ∂t = b x n ∂ ∂x  x n ∂w ∂x  + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1357 T10.3.3. Systems of the Form Δu = F (u, w), Δw = G(u, w) . . . . . . . . . . . . . . . . . . 1364 T10.3.4. Systems of the Form ∂ 2 u ∂t 2 = a x n ∂ ∂x  x n ∂u ∂x  + F (u, w), ∂ 2 w ∂t 2 = b x n ∂ ∂x  x n ∂w ∂x  + G(u, w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368 T10.3.5. Other Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373 T10.4. Systems of General Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374 T10.4.1. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374 T10.4.2. Nonlinear Systems of Two Equations Involving the First Derivatives in t . . 1374 T10.4.3. Nonlinear Systems of Two Equations Involving the Second Derivatives in t 1378 T10.4.4. Nonlinear Systems of Many Equations Involving the First Derivatives in t . 1381 References for Chapter T10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1382 T11. Integral Equations 1385 T11.1. Linear Equations of the First Kind with Variable Limit of Integration . . . . . . . . . . . . . 1385 T11.2. Linear Equations of the Second Kind with Variable Limit of Integration . . . . . . . . . . . 1391 T11.3. Linear Equations of the First Kind with Constant Limits of Integration . . . . . . . . . . . . 1396 T11.4. Linear Equations of the Second Kind with Constant Limits of Integration . . . . . . . . . 1401 References for Chapter T11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406 T12. Functional Equations 1409 T12.1. Linear Functional Equations in One Independent Variable . . . . . . . . . . . . . . . . . . . . . . 1409 T12.1.1. Linear Difference and Functional Equations Involving Unknown Function with Two Different Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1409 T12.1.2. Other Linear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1421 T12.2. Nonlinear Functional Equations in One Independent Variable . . . . . . . . . . . . . . . . . . . 1428 T12.2.1. Functional Equations with Quadratic Nonlinearity . . . . . . . . . . . . . . . . . . . . 1428 T12.2.2. Functional Equations with Power Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . 1433 T12.2.3. Nonlinear Functional Equation of General Form . . . . . . . . . . . . . . . . . . . . . 1434 T12.3. Functional Equations in Several Independent Variables . . . . . . . . . . . . . . . . . . . . . . . . 1438 T12.3.1. Linear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438 T12.3.2. Nonlinear Functional Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1443 References for Chapter T12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1450 Supplement. Some Useful Electronic Mathematical Resources 1451 Index 1453 AUTHORS Andrei D. Polyanin, D.Sc., Ph.D., is a well-known scientist of broad interests who is active in various areas of mathe- matics, mechanics, and chemical engineering sciences. He is one of the most prominent authors in the field of reference literature on mathematics and physics. Professor Polyanin graduated with honors from the De- partment of Mechanics and Mathematics of Moscow State University in 1974. He received his Ph.D. in 1981 and his D.Sc. in 1986 from the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, Professor Polyanin has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences; he is also Professor of Mathematics at Bauman Moscow State Technical University. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and of the Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation. Professor Polyanin has made important contributions to exact and approximate analytical methods in the theory of differential equations, mathematical physics, integral equations, engineering mathematics, theory of heat and mass transfer, and chemical hydrodynamics. He has obtained exact solutions for several thousand ordinary differential, partial differen- tial, and integral equations. Professor Polyanin is an author of more than 30 books in English, Russian, German, and Bulgarian as well as more than 120 research papers and three patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995 and 2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, 2002; A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equation, Chapman & Hall/CRC Press, 2004. Professor Polyanin is editor of the book series Differential and Integral Equations and Their Applications, Chapman & Hall/CRC Press, London/Boca Raton, and Physical and Mathematical Reference Literature, Fizmatlit, Moscow. He is also Editor-in-Chief of the international scientific-educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru), which is visited by over 1000 users a day worldwide. Professor Polyanin is a member of the Editorial Board of the journal Theoretical Foundations of Chemical Engineering. In 1991, Professor Polyanin was awarded a Chaplygin Prize of the Russian Academy of Sciences for his research in mechanics. In 2001, he received an award from the Ministry of Education of the Russian Federation. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia Home page: http://eqworld.ipmnet.ru/polyanin-ew.htm xxv xxvi AUTHORS Alexander V. Manzhirov, D.Sc., Ph.D., is a noted scientist in the fields of mechanics and applied mathematics, integral equations, and their applications. After graduating with honors from the Department of Mechanics and Mathematics of Rostov State University in 1979, Professor Manzhirov attended postgraduate courses at Moscow Institute of Civil Engineering. He received his Ph.D. in 1983 from Moscow Institute of Electronic Engi- neering Industry and his D.Sc. in 1993 from the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1983, Professor Manzhirov has been working at the Institute for Problems in Mechanics of the Russian Academy of Sciences, where he is currently head of the Laboratory for Modeling in Solid Mechanics. Professor Manzhirov is also head of a branch of the Department of Applied Mathematics at Bauman Moscow State Technical University, professor of mathematics at Moscow State University of Engineering and Computer Science, vice-chairman of Mathematics and Mechanics Expert Council of the Higher Certification Committee of the Russian Federation, executive secretary of Solid Mechanics Scientific Council of the Russian Academy of Sciences, and an expert in mathematics, mechanics, and computer science of the Russian Foundation for Basic Research. He is a member of the Russian National Committee on Theoretical and Applied Mechanics and the European Mechanics Society (EUROMECH), and a member of the editorial board of the journal Mechanics of Solids and the international scientific-educational Website EqWorld—The World of Mathematical Equations (http://eqworld.ipmnet.ru). Professor Manzhirov has made important contributions to new mathematical methods for solving problems in the fields of integral equations and their applications, mechanics of growing solids, contact mechanics, tribology, viscoelasticity, and creep theory. He is the au- thor of ten books (including Contact Problems in Mechanics of Growing Solids [in Russian], Nauka, Moscow, 1991; Handbook of Integral Equations, CRC Press, Boca Raton, 1998; Handbuch der Integralgleichungen: Exacte L ¨ osungen, Spektrum Akad. Verlag, Heidelberg, 1999; Contact Problems in the Theory of Creep [in Russian], National Academy of Sciences of Armenia, Erevan, 1999), more than 70 research papers, and two patents. Professor Manzhirov is a winner of the First Competition of the Science Support Foundation 2001, Moscow. Address: Institute for Problems in Mechanics, Vernadsky Ave. 101 Bldg 1, 119526 Moscow, Russia Home page: http://eqworld.ipmnet.ru/en/board/manzhirov.htm PREFACE This book can be viewed as a reasonably comprehensive compendium of mathematical definitions, formulas, and theorems intended for researchers, university teachers, engineers, and students of various backgrounds in mathematics. The absence of proofs and a concise presentation has permitted combining a substantial amount of reference material in a single volume. When selecting the material, the authors have given a pronounced preference to practical aspects, namely, to formulas, methods, equations, and solutions that are most frequently used in scientific and engineering applications. Hence some abstract concepts and their corollaries are not contained in this book. • This book contains chapters on arithmetics, elementary geometry, analytic geometry, algebra, differential and integral calculus, differential geometry, elementary and special functions, functions of one complex variable, calculus of variations, probability theory, mathematical statistics, etc. Special attention is paid to formulas (exact, asymptotical, and approximate), functions, methods, equations, solutions, and transformations that are of frequent use in various areas of physics, mechanics, and engineering sciences. • The main distinction of this reference book from other general (nonspecialized) math- ematical reference books is a significantly wider and more detailed description of methods for solving equations and obtaining their exact solutions for various classes of mathematical equations (ordinary differential equations, partial differential equations, integral equations, difference equations, etc.) that underlie mathematical modeling of numerous phenomena and processes in science and technology. In addition to well-known methods, some new methods that have been developing intensively in recent years are described. • For the convenience of a wider audience with different mathematical backgrounds, the authors tried to avoid special terminology whenever possible. Therefore, some of the methods and theorems are outlined in a schematic and somewhat simplified manner, which is sufficient for them to be used successfully in most cases. Many sections were written so that they could be read independently. The material within subsections is arranged in increasing order of complexity. This allows the reader to get to the heart of the matter quickly. The material in the first part of the reference book can be roughly categorized into the following three groups according to meaning: 1. The main text containing a concise, coherent survey of the most important definitions, formulas, equations, methods, and theorems. 2. Numerous specific examples clarifying the essence of the topics and methods for solving problems and equations. 3. Discussion of additional issues of interest, given in the form of remarks in small print. For the reader’s convenience, several long mathematical tables—finite sums, series, indefinite and definite integrals, direct and inverse integral transforms (Laplace, Mellin, and Fourier transforms), and exact solutions of differential, integral, functional, and other mathematical equations—which contain a large amount of information, are presented in the second part of the book. This handbook consists of chapters, sections, subsections, and paragraphs (the titles of the latter are not included in the table of contents). Figures and tables are numbered sep- arately in each section, while formulas (equations) and examples are numbered separately in each subsection. When citing a formula, we use notation like (3.1.2.5), which means xxvii . Press, 1995 and 2003; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists,. V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, 2002; and A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equation,. compendium of mathematical definitions, formulas, and theorems intended for researchers, university teachers, engineers, and students of various backgrounds in mathematics. The absence of proofs and