402 FUNCTIONS OF COMPLEX VARIABLE of D with its boundary is called a closed domain and is denoted by D.Thepositive sense of the boundary is defined to be the sense for which the domain lies to the left of the boundary. The boundary of a domain can consist of finitely many closed curves, segments, and points; the curves and cuts are assumed to be piecewise smooth. The simplest examples of domains are neighborhoods of points on the complex plane. A neighborhood of a point a on the complex plane is understood as the set of points z such that |z – a| < R, i.e., the interior of the disk of radius R > 0 centered at the point a.The extended complex plane is obtained by augmenting the complex plane with the fictitious point at infinity.Aneighborhood of the point at infinity is understood as the set of points z such that |z| > R (including the point at infi nity itself). If to each point z of a domain D there corresponds a point w (resp., a set of points w), then one says that there is a single-valued (resp., multivalued) function w = f(z)defined on the domain D.Ifwesetz = x + iy and w = u + iv,thendefining a function w = f(z ) of the complex variable is equivalent to defining two functions Re f = u = u(x, y)and Im f = v = v(x, y) of two real variables. If the function w = f(z) is single-valued on D and the images of distinct points of D are distinct, then the mapping determined by this function is said to be schlicht. The notions of boundedness, limit, and continuity for single-valued functions of the complex variable do not differ from the corresponding notions for functions of two real variables. 10.1.2-2. Differentiability and analyticity. Let a single-valued function w = f (z)bedefined in a neighborhood of a point z.Ifthere exists a limit lim h→0 f(z + h)–f(z) h = f z (z), (10.1.2.1) then the function w = f(z)issaidtobedifferentiable at the point z and f z (z) is called its derivative at the point z. Cauchy–Riemann conditions. If the functions u(x, y)=Ref(z)andv(x, y)=Imf(z) are differentiable at a point (x, y), then the Cauchy–Riemann conditions ∂u ∂x = ∂v ∂y , ∂u ∂y =– ∂v ∂x (10.1.2.2) are necessary andsufficient for the function w =f (z) to bedifferentiable at the point z =x+iy. If the function w = f (z) is differentiable, then w z = u x + iv x = v y – iu y = u x + iu y = v y + iv x ,(10.1.2.3) where the subscripts x and y indicate the corresponding partial derivatives. Remark. The Cauchy–Riemann conditions are sometimes also called the d’Alembert–Euler conditions. The rules for algebraic operations on the derivatives and for calculating the derivative of the composite function and the inverse function (if it exists) have exactly the same form as in the real case: 1. αf 1 (z) βf 2 (z) z = α[f 1 (z)] z β[f 2 (z)] z ,whereα and β are arbitrary complex constants. 2. f 1 (z)f 2 (z) z =[f 1 (z)] z f 2 (z)+f 1 (z)[f 2 (z)] z . 3. f 1 (z) f 2 (z) z = [f 1 (z)] z f 2 (z)–f 1 (z)[f 2 (z)] z f 2 2 (z) (f 2 (z) 0). 4. If a function w = f(z) is differentiable at a point z and a function W = F(w)is differentiable at the point w = f(z), then the composite function W = F(f(z)) is differentiable at the point z and W z =[F (f(z))] z = F f (f)f z (z). 10.1. BASIC NOTIONS 403 5. Ifa function w =f(z) is differentiable ata point z and theinverse function z =g(w)≡f –1 (w) exists and is differentiable at the point w and satisfies [f –1 (w)] w ≠ 0,then f z (z)= 1 [f –1 (w)] w . A single-valued function differentiable in some neighborhood of a point z 0 is said to be analytic (regular, holomorphic) at this point. A function w = f(z) is analytic at a point z 0 if and only if it can be represented by a power series f(z)= ∞ k=0 c k (z – z 0 ) k (10.1.2.4) converging in some neighborhood of z 0 . A function analytic at each point of the domain D is said to be analytic in D. A function w = f (z)issaidtobeanalytic at the point at infinity if the function F (z)=f(1/z) is analytic at the point z = 0. In this case, f z (∞)=(–z 2 F z ) z=0 by definition. A function w = f(z) is analytic at the point at infinity if and only if this function can be represented by a power series f(z)= ∞ k=0 b k z –k (10.1.2.5) converging for sufficiently large |z|. If a function w = f (z) is analytic at a point z 0 and f z (z 0 ) ≠ 0,thenf(z) has an analytic inverse function z(w)defined in a neighborhood of the point w 0 = f(z 0 ). If a function w = f (z) is analytic at a point z 0 and the function W = F (w) is analytic at the point w 0 = f (z 0 ), then the composite function W = F [f (z)] is analytic at the point z 0 .Ifa function is analytic in a domain D and continuous in D, then its value at any interior point of the domain is uniquely determined by its values on the boundary of the domain. The analyticity of a function at a point implies the existence and analyticity of its derivatives of arbitrary order at this point. M AXIMUM MODULUS PRINCIPLE. If a function w = f (z) that is not identically constant is analytic in a domain D and continuous in D , then its modulus cannot attain a maximum at an interior point of D . LIOUVILLE’S THEOREM. If a function w = f (z) is analytic and bounded in the entire complex plane, then it is constant. Remark. The Liouville theorem can be stated in the following form: if a function w = f(z) is analytic in the extended complex plane, then it is constant. Geometric meaning of the absolute value of the derivative. Suppose that a function w = f (z) is analytic at a point z 0 and f z (z 0 ) ≠ 0. Then the value |f z (z 0 )| determines the dilatation (similarity) coefficient at the point z 0 under the mapping w = f(z). The value |f z (z 0 )| is called the dilatation ratio if |f z (z 0 )| > 1 and the contraction ratio if |f z (z 0 )| < 1. Geometric meaning of the argument of the derivative. The argument of the derivative f z (z 0 ) is equal to the angle by which the tangent at the point z 0 to any curve passing through z 0 should be rotated to give the tangent to the image of the curve at the point w 0 = f(z 0 ). For ϕ =argf z (z)>0, the rotation is anticlockwise, and for ϕ =argf z (z)<0, the rotation is clockwise. Single-valued functions, as well as single-valued branches of multi-valued functions, are analytic everywhere on the domains where they are defined. It follows from (10.1.2.2) 404 FUNCTIONS OF COMPLEX VARIABLE that the real and imaginary parts u(x, y)andv(x, y) of a function analytic in a domain are harmonic in this domain, i.e., satisfy the Laplace equation Δf = f xx + f yy = 0 (10.1.2.6) in this domain. Remark. If u(x, y)andv(x, y) are two arbitrary harmonic functions, then the function f(z)=u(x, y)+ iv(x, y) is not necessarily analytic, since for the analyticity of f(z) the functions u(x, y)andv(x, y) must satisfy the Cauchy–Riemann conditions. Example 1. The function w = z 2 is analytic. Indeed, since z = x + iy,wehavew =(x + iy) 2 = x 2 – y 2 + i2xy, u(x, y)=x 2 – y 2 ,andv(x, y)=2xy.The Cauchy–Riemann conditions u x = v y = 2x, u y = v x =–2y are satisfied at all points of the complex plane, and the function w = z 2 is analytic. Example 2. The function w = ¯z is not analytic. Indeed, since z = x + iy,wehavew = x – iy, u(x, y)=x, v(x, y)=–y. The Cauchy–Riemann conditions are not satisfied, u x = 1 ≠ –1 = v y , u y = v x = 0, and the function w = ¯z is not analytic. 10.1.2-3. Elementary functions. 1 ◦ .Thefunctionsw =z n and w= n √ z for positive integer n are definedinParagraph10.1.1-2. The function w = z n (10.1.2.7) is single-valued. It is schlicht in the sectors 2πk/n < ϕ < 2π(k + 1)/n, k = 0, 1, 2, , each of which is transformed by the mapping w = z n into the plane w with a cut on the positive semiaxis. The function w = n √ z (10.1.2.8) is an n-valued function for z ≠ 0, and its value is determined by the value of the argument chosen for the point z. If a closed curve C does not surround the point z = 0, then, as the point z goes around the entire curve C, the point w = n √ z for a chosen value of the root also moves along a closed curve and returns to the initial value of the argument. But if the curve C surrounds the origin, then, as the point z goes around the entire curve C in the positive sense, the argument of z increases by 2π and the corresponding point w = n √ z does not return to the initial position. It will return there only after the point z goes n times around the entire curve C. If a domain D does not contain a closed curve surrounding the point z = 0, then one can singe out n continuous single-valued functions, each of which takes only one of the values w = n √ z; these functions are called the branches of the multi-valued function w = n √ z. One cannot single n separate branches of the function w = n √ z in any neighborhood of the point z = 0; accordingly, the point z = 0 is called a branch point of this function. 2 ◦ .TheZhukovskii function w = 1 2 z + 1 z (10.1.2.9) is defined and single-valued for all z ≠ 0; it is schlicht in any domain that does not simultaneously contain any points z 1 and z 2 such that z 1 z 2 = 1. 10.1. BASIC NOTIONS 405 3 ◦ .Theexponential function w = e z is defined by the formula w = e z = e x+iy = e x (cos y + i sin y). (10.1.2.10) The function w = e z is analytic everywhere. For the exponential function, the usual differentiation rule is preserved: (e z ) z = e z . The basic property of the exponential function (addition theorem) is also preserved: e z 1 e z 2 = e z 1 +z 2 . For x = 0 and y = ϕ, the definition of the exponential function implies the Euler formula e iϕ =cosϕ + i sin ϕ, which permits one to write any complex number with modulus r and argument ϕ in the exponential form z = r(cos ϕ + i sin ϕ)=re iϕ .(10.1.2.11) The exponential function is 2π-periodic, and the mapping w = e z is schlicht in the strip 0 ≤ y < 2π. 4 ◦ .Thelogarithm is defined as the inverse of the exponential function: if e w = z,then w =Lnz.(10.1.2.12) This function is defined for z ≠ 0. The logarithm satisfies the following relations: Ln z 1 +Lnz 2 =Ln(z 1 z 2 ), Ln z 1 –Lnz 2 =Ln z 1 z 2 , Ln(z n )=n Ln z,Ln n √ z = 1 n Ln z. The exponential form of complex numbers readily shows that the logarithm is infinite- valued: Ln z =ln|z| + i Arg z =ln|z| + i arg z + 2πki, k = 0, 1, 2, (10.1.2.13) The value ln z =ln|z| + i arg z is taken to be the principal value of this function. Just as with the function w = n √ z, we see that if the point z = 0 is surrounded by a closed curve C, then the point w =Lnz does not return to its initial position after z goes around C in the positive sense, since the argument of w increases by 2πi. Thus if a domain D does not contain a closed curve surrounding the point z = 0,theninD one can single out infinitely many continuous and single-valued branches of the multi-valued function w =Lnz;the differences between the values of these branches at each point of the domain have the form 2πki,wherek is an integer. This cannot be done in an arbitrary neighborhood of the point z = 0, and this point is called a branch point of the logarithm. 5 ◦ . Trigonometric functions are defined in terms of the exponential function as follows: cos z = e iz + e –iz 2 ,sinz = e iz – e –iz 2i , tan z = sin z cos z =–i e iz – e –iz e iz + e –iz ,cotz = cos z sin z = i e iz + e –iz e iz – e –iz . (10.1.2.14) 406 FUNCTIONS OF COMPLEX VARIABLE Properties of the functions cos z and sin z: 1. They are analytic for any z. 2. The usual differentiation rules are valid: (sin z) z =cosz,(cosz) z =sinz. 3. They are periodic with real period T = 2π. 4. sin z is an odd function, and cos z is an even function. 5. In the complex domain, they are unbounded. 6. The usual trigonometric relations hold: cos 2 z +sin 2 z = 1,cos2z =cos 2 z –sin 2 z,etc. The function tan z is analytic everywhere except for the points z k = π 2 + kπ, k = 0, 1, 2, , and the function cot z is analytic everywhere except for the points z k = kπ, k = 0, 1, 2, The functions tan z and cot z are periodic with real period T = π. 6 ◦ . Hyperbolic functions are defined by the formulas cosh z = e z + e –z 2 ,sinhz = e z – e –z 2 , tanh z = sinh z cosh z = e z – e –z e z + e –z ,cothz = cosh z sinh z = e z + e –z e z – e –z . (10.1.2.15) For real values of the argument, each of these functions coincides with the corresponding real function. Hyperbolic and trigonometric functions are related by the formulas cosh z =cosiz,sinhz =–i sin iz,tanhz =–i tan iz,cothz = i cot iz. 7 ◦ . Inverse trigonometric and hyperbolic functions are expressed via the logarithm and hence are infinite-valued: Arccos z =–i Ln(z + √ z 2 – 1), Arcsin z =–i Ln(iz + √ 1 – z 2 ), Arctan z =– i 2 Ln 1 + iz 1 – iz , Arccot z =– i 2 Ln z + i z – i , arccosh z =Ln(z + √ z 2 – 1), arcsinh z =Ln(z + √ z 2 – 1), arctanh z = 1 2 Ln 1 + z 1 – z , arccoth z = 1 2 Ln z + 1 z – 1 . (10.1.2.16) The principal value of each of these functions is obtained by choosing the principal value of the corresponding logarithmic function. 8 ◦ .Thepower w = z γ is defined by the relation z γ = e γ Ln z ,(10.1.2.17) where γ = α + iβ is an arbitrary complex number. Substituting z = re iϕ into (10.1.2.17), we obtain z γ = e α ln r–β(ϕ+2kπ) e iα(ϕ+2kπ)+iβ ln r , k = 0, 1, 2, (10.1.2.18) It follows from relation (10.1.2.18) that the function w = z γ has infinitely many values for β ≠ 0. 10.1. BASIC NOTIONS 407 9 ◦ .Thegeneral exponential function is defined by the formula w = γ z = e z Lnγ = e z ln |γ| e ziArg γ ,(10.1.2.19) where γ = α + iβ is an arbitrary nonzero complex number. The function (10.1.2.19) is a set of separate mutually independent single-valued functions that differ from one another by the factors e 2kπiz , k = 0, 1, 2, Example 3. Let us calculate the values of some elementary functions at specific points: 1. cos 2i = 1 2 (e 2ii + e –2ii )= 1 2 (e 2 + e –2 )=cosh2 ≈ 3.7622. 2. ln(–2)=ln2 + iπ,since| – 2| = 2 and the principal value of the argument is equal to π. 3. Ln(–2) is calculated by formula (10.1.2.13): Ln(–2)=ln2 + iπ + 2πki =ln2 +(1 + 2k)iπ (k = 0, 1, 2, ). 4. i i = e i Ln i = e i(iπ/2+2πk) = e –π/2–2πk (k = 0, 1, 2, ). The main elementary functions w = f(z)=u(x, y)+iv(x, y)ofthecomplexvariable z = x + iy are given in Table 10.1. 10.1.2-4. Integration of function of complex variable. Suppose that an oriented curve C connecting points z = a and z = b isgivenonthecomplex plane and a function w = f(z)ofthecomplexvariableisdefined on the curve. We divide the curve C into n parts, a = z 0 , z 1 , , z n–1 , z n = b, arbitrarily choose ξ k [z k , z k+1 ], and compose the integral sum n–1 k=0 f(ξ k )(z k+1 – z k ). If there exists a limit of this sum as max |z k+1 – z k | → 0, independent of the construction of the partition and the choice of points ξ k , then this limit is called the integral of the function w = f (z) over the curve C and is denoted by C f(z) dz.(10.1.2.20) Properties of the integral of a function of a complex variable: 1. If α, β are arbitrary constants, then C [αf(z)+βg(z)] dz = α C f(z) dz +β C g(z) dz. 2. If C is the same curve as C but with the opposite sense, then C f(z) dz =– C f(z) dz. 3. If C = C 1 ∪···∪C n ,then C f(z) dz = C 1 f(z) dz + + C n f(z) dz. 4. If |f(z)| ≤ M at all points of the curve C, then the following estimate of the absolute value of the integral holds: C f(z) dz ≤ Ml,wherel is the length of the curve C. If C is a piecewise smooth curve and f (z) is bounded and piecewise continuous, then the integral (10.1.2.20) exists. If z = x + iy and w = u(x, y)+iv(x, y), then the computation of the integral (10.1.2.20) is reduced to finding two ordinary curvilinear integrals: C f(z) dz = C u(x, y) dx – v(x, y) dy + i C v(x, y) dx + u(x, y) dy.(10.1.2.21) Remark. Formula (10.1.2.21) can be rewritten in a form convenient for memorizing: C f(z) dz = C (u + iv)(dx + idy). 408 FUNCTIONS OF COMPLEX VARIABLE TABLE 10.1 Main elementary functions w = f(z)=u(x, y)+iv(x, y) of complex variable z = x + iy No. Complex function w = f(z) Algebraic form f(z)=u(x, y)+iv(x, y) Zeros of nth order Singularities 1 z x + iy z = 0, n = 1 z = ∞ is a first-order pole 2 z 2 x 2 – y 2 + i 2xy z = 0, n = 2 z = ∞ is a second-order pole 3 1 z–(x 0 +iy 0 ) (x 0 , y 0 are real numbers) x–x 0 (x–x 0 ) 2 +(y–y 0 ) 2 + i –(y–y 0 ) (x–x 0 ) 2 +(y–y 0 ) 2 z = ∞, n = 1 z = x 0 + iy 0 is a first-order pole 4 1 z 2 x 2 – y 2 (x 2 + y 2 ) 2 + i –2xy (x 2 + y 2 ) 2 z = ∞, n = 2 z = 0 is a second-order pole 5 √ z x+ x 2 +y 2 2 1/2 +i –x+ x 2 +y 2 2 1/2 z = 0 is a branch point z = 0 is a first-order branch point z = ∞ is a first-order branch point 6 e z e x cos y + ie x sin y — z = ∞ is an essential singular point 7 Ln z ln |z| + i(arg z + 2kπ), k = 0, 1, 2, z = 1, n = 1 (for the branch corresponding to k = 0) Logarithmic branch points for z = 0, z = ∞ 8 sin z sin x cosh y + i cos x sinh y z = πk, n = 1 (k = 0, 1, 2, ) z = ∞ is an essential singular point 9 cos z cos x cosh y + i(– sin x sinh y) z = 1 2 π + πk, n = 1 (k = 0, 1, 2, ) z = ∞ is an essential singular point 10 tan z sin 2x cos 2x +cosh2y + i sinh 2y cos 2x +cosh2y z = πk, n = 1 (k = 0, 1, 2, ) z = 1 2 π + πk (k = 0, 1, 2, ) are first-order poles If the curve C is given by the parametric equations x = x(t), y = y(t)(t 1 ≤ t ≤ t 2 ), then C f(z) dz = t 2 t 1 f(z(t))z t (t) dt,(10.1.2.22) where z = z(t)=x(t)+iy(t) is the complex parametric equation of the curve C. If f(z) is an analytic function in a simply connected domain D containing the points z = a and z = b, then the Newton–Leibniz formula holds: b a f(z) dz = F (b)–F (a), (10.1.2.23) where F (z) is a primitive of the function f (z), i.e., F z (z)=f(z) in the domain D. . there exists a limit of this sum as max |z k+1 – z k | → 0, independent of the construction of the partition and the choice of points ξ k , then this limit is called the integral of the function w. boundary of a domain can consist of finitely many closed curves, segments, and points; the curves and cuts are assumed to be piecewise smooth. The simplest examples of domains are neighborhoods of points. schlicht. The notions of boundedness, limit, and continuity for single-valued functions of the complex variable do not differ from the corresponding notions for functions of two real variables. 10.1.2-2.