416 FUNCTIONS OF COMPLEX VARIABLE 4. If f(z) is the quotient of two analytic functions, f(z)= ϕ(z) ψ(z) , in a neighborhood of a point a and ϕ(a) ≠ 0, ψ(a)=0,butψ z (a) ≠ 0 (i.e., a is a simple pole of f (z)), then res f(a)= ϕ(a) ψ z (a) .(10.1.2.48) 5. If a is an essential singularity of f(z), then to obtain res f(a), one has to find the coefficient c –1 in the Laurent expansion of f (z) in a neighborhood of a. A function f (z) is said to be continuous on the boundary C of the domain D if for each boundary point z 0 there exists a limit lim z→z 0 f(z)=f(z 0 )asz → z 0 , z D. C AUCHY’S RESIDUE THEOREM. Let f(z) be a function continuous on the boundary C of a domain D and analytic in the interior of D everywhere except for finitely many points a 1 , , a n .Then C f(z) dz = 2πi n k=1 res f(a k ), (10.1.2.49) where the integral is taken in the positive sense of C . The logarithmic residue of a function f (z) at a point a is by defi nition the residue of its logarithmic derivative ln f(z) z = f z (z) f(z) . T HEOREM. The logarithmic derivative f z (z)/f(z) has first-order poles at the zeros and poles of f(z) . Moreover, the logarithmic residue of f(z) at a zero or a pole of f(z) is equal to the order of the zero or minus the order of the pole, respectively. The residue of a function f (z)atinfinity is defined as res f(∞)= 1 2πi Γ f(z) dz,(10.1.2.50) where Γ is a circle of sufficiently large radius |z| = ρ and the integral is taken in the clockwise sense (so that the neighborhood of the point z = ∞ remains to the left of the contour, just as in the case of a finite point). The residue of f(z)atinfinity is equal to minus the coefficient of z –1 in the Laurent expansion of f(z) in a neighborhood of the point z = ∞, res f(∞)=–c –1 .(10.1.2.51) Note that res f(∞) = lim z→∞ [–zf(z)], (10.1.2.52) provided that this limit exists. T HEOREM. If a function f(z) has finitely many singular points a 1 , , a n in the extended complex plane, then the sum of all its residues, including the residue at infinity, is zero: res f(∞)+ n k=1 res f(a k )=0.(10.1.2.53) 10.1. BASIC NOTIONS 417 Example 10. Let us calculate the integral C ln(z + 2) z 2 dz, where C is the circle |z| = 1 2 . In the disk |z| ≤ 1 2 , there is only one singular point of the integrand, z = 0, which is a second-order pole. The residue of f(z)atz = 0 is calculated by the formula (10.1.2.46) res f (0) = lim z→0 z 2 ln(z + 2) z 2 z = lim z→0 [ln(z + 2)] z = lim z→0 1 z + 2 = 1 2 . Using formula (10.1.2.44), we obtain 1 2 = 1 2πi C ln(z + 2) z 2 dz, C ln(z + 2) z 2 dz = πi. 10.1.2-8. Calculation of definite integrals. Suppose that we need to calculate the integral of a real function f(x) over a (finite or infinite) interval (a, b). Let us supplement the interval (a, b) with a curve Γ that, together with (a, b), bounds a domain D, and then analytically continue the function f(x)into D. Then the residue theorem can be applied to this analytic continuation of f (z), and by this theorem, b a f(x) dx + Γ f(z) dz = 2πiΛ,(10.1.2.54) where Λ is the sum of residues of f(z)inD.If Γ f(z) dz can be calculated or expressed in terms of the desired integral b a f(x) dx, then the problem will be solved. When calculating integrals of the form ∞ –∞ f(x) dx, one should apply (10.1.2.49) to the contour C that consists of the interval (–R, R) of the real axis and the arc C R of the circle |z| = R in the upper half-plane. Sometimes it is only possible to find the limit as R →∞ of the integral over the contour C R rather than to calculate it, and often it turns out that the limit of this integral is equal to zero. The integral over the curve Γ can be estimated using the following lemmas. J ORDAN LEMMA. If a function g(z) tends to zero uniformly with respect to arg z along a sequence of circular arcs C R n : |z| = R n , Im z >–a (where R n →∞ and a is fixed), then lim n→∞ C R n g(z)e imz dz = 0 (10.1.2.55) for each positive number m . If a function f(z) is analytic for |z| > R 0 and zf(z) → 0 as |z| →∞for y ≥ 0,then lim R→∞ C R f(z) dz = 0,(10.1.2.56) where C R is the arc of the circle |z| = R in the upper half-plane. 418 FUNCTIONS OF COMPLEX VARIABLE X R ai R Y C R Figure 10.5. The contour to calculate the Laplace integral. Example 11 (Laplace integral). To calculate the integral ∞ 0 cos x x 2 + a 2 dx, one uses the auxiliary function f(z)= e iz z 2 + a 2 = g(z)e iz , g(z)= 1 z 2 + a 2 and the contour shown in Fig. 10.5. Since g(z) satisfies the inequality |g(z)| <(R 2 – a 2 ) –1 on C R , it follows that this function uniformly tends to zero as R →∞, and by the Jordan lemma we obtain C R f(z) dz = C R g(z)e iz dz → 0 as R →∞. By the residue theorem, R –R e ix x 2 + a 2 dx + C R f(z) dz = 2πi e –a 2ai for each R > 0. (The residue at the singular point z = ai of the function f(z), which is a first-order pole and which is the only singular point of this function lying inside the contour, can be calculated by formula (10.1.2.48).) In the limit as R →∞, we obtain ∞ –∞ e ix x 2 + a 2 dx = π ae a . Separating the real part and using the fact that the function is even, we obtain ∞ 0 cos x x 2 + a 2 dx = π 2ae a . 10.1.2-9. Analytic continuation. Let two domains D 1 and D 2 have a common part γ of the boundary, and let single-valued analytic functions f 1 (z)andf 2 (z), respectively, be given in these domains. The function f 2 (z) is called a direct analytic continuation of f 1 (z) into the domain D 2 if there exists a function f(z) analytic in the domain D 1 ∪γ ∪D 2 and satisfying the condition f(z)= f 1 (z)forz D 1 , f 2 (z)forz D 2 . (10.1.2.57) If such a continuation is possible, then the function f(z) is uniquely determined. If the domains are simply connected and the functions f 1 (z)andf 2 (z) are continuous in D 1 ∪ γ and D 2 ∪ γ, respectively, and coincide on γ,thenf 2 (z) is the direct analytic continuation of f 1 (z) into the domain D 2 . In addition, suppose that the domains D 1 and D 2 are allowed to have common interior points. A function f 2 (z) is called a direct analytic continuation of f 1 (z) through γ if f 1 (z)andf 2 (z) are continuous in D 1 ∪ γ and D 2 ∪ γ, respectively, and their values on γ coincide. At the common interior points of D 1 and D 2 , the function determined by relation (10.1.2.57) can be double-valued. 10.2. MAIN APPLICATIONS 419 10.2. Main Applications 10.2.1. Conformal Mappings 10.2.1-1. Generalities. A one-to-one mapping w = f(z)=u(x, y)+iv(x, y)(10.2.1.1) of a domain D onto a domain D ∗ is said to be conformal if the principal linear part of this mapping at any point of D is an orthogonal orientation-preserving transformation. Main properties of conformal mappings: 1. Circular property. A conformal mapping takes infinitesimal circles to infinitesimal circles (up to higher-order infinitesimals). 2. Angle preservation property. A conformal mapping preserves the angles between intersecting curves at points of intersection. T HEOREM. A function w = f(z) is a conformal mapping of a domain D if and only if it is analytic and schlicht in D and the derivative f z (z) vanishes nowhere in D . The main problem in the theory of conformal mappings is as follows: given domains D and D ∗ , construct a function that gives a conformal mapping of one of the domains onto the other. T HE MAIN THEOREM OF THE THEORY OF CONFORMAL MAPPINGS (RIEMANN THEOREM). For any simply connected domains D and D ∗ (with boundaries consisting of more than a single point), any points z 0 D and w 0 D ∗ , and any real number α 0 , there exists a unique conformal mapping w = f(z) of D onto D ∗ such that f(z 0 )=w 0 ,argf z (z 0 )=α 0 . 10.2.1-2. Boundary correspondence. On the boundary C of a domain D, let us introduce a real arc length parameter s reckoned from some point of C,sothatζ = ζ(s)onC.Iff (z) is a continuous function in the closed domain D, then on the boundary C one can set f(ζ)=f[ζ(s)] = ϕ(s). The function ϕ(s) is called the boundary function for f(z). T HEOREM ON THE BOUNDARY CORRESPONDENCE. Suppose that a function w = f (z) specifies a conformal mapping between domains D and D ∗ . Then the following assertions hold. 1. If the boundary of D ∗ does not have infinite branches, then f(z) is continuous on the boundary of D and the boundary function w = f (ζ)=ϕ(s) is a continuous one-to-one correspondence between the boundaries of the domains D and D ∗ . 2. If the boundaries of D and D ∗ do not contain infinite branches and have a continuous curvature at each point, then the boundary function ϕ(s) is continuously differentiable. 420 FUNCTIONS OF COMPLEX VARIABLE BOUNDARY CORRESPONDENCE PRINCIPLE. Let D and D ∗ be two simply connected domains with boundaries C and C ∗ , and let the domain D ∗ be bounded. Suppose that a function w = f(z) satisfies the following conditions: 1. It is analytic in D and continuous in D . If the point at infinity lies in the interior of the domain D ∗ , then the boundary correspondence principle remains valid provided that w = f(z) is continuous in D and analytic in D everywhere except for an interior point z 0 , at which this function has a simple pole. 2. It is a one-to-one sense-preserving mapping of C onto C ∗ . Then f(z) is a (schlicht) conformal mapping of D onto D ∗ . Example 1. The exponential function w = e z maps a) the strip between the straight lines y = k(x – a 1 )andy = k(x – a 2 ) onto the strip lying between the logarithmic spirals (Fig. 10.6). (If k(a 2 – a 1 )=2π, then the spirals coincide, and we obtain a mapping onto the plane with the spiral cut; for k(a 2 – a 1 )>2π, the mapping is not schlicht); b) the strip 0 <Imz < π onto the upper half-plane (Fig. 10.7); here the point πi is taken to the point –1,and the point 0 is taken to the point 1; c) the half-strip 0 <Imz < π,Rez < 0, onto the half-disk |w| < 1,Imw > 0 (Fig. 10.8); d) a rectangle onto a half-annulus (Fig. 10.9). X U Y V a aa ee a 1 12 2 Figure 10.6. The exponential function w = e z maps the strip between the straight lines onto the strip lying between the logarithmic spirals. X πi 11 U Y V Figure 10.7. The exponential function w = e z maps the strip 0 <Imz < π onto the upper half-plane. X U 11 Y V πi Figure 10.8. The exponential function w = e z maps the half-strip 0 <Imz < π,Rez < 0, onto the half-disk. 10.2. MAIN APPLICATIONS 421 X U Y V πi aa 12 aa ee 12 Figure 10.9. The exponential function w = e z maps a rectangle onto a half-annulus. Example 2. The function w = z 2 maps the interior of a circle onto the interior of a cardioid (Fig. 10.10). The circle given in polar coordinates by the equation r =cosϕ is taken to the cardioid ρ = 1 2 (1 +cosθ), where θ = 2ϕ. X O 11 O U Y V Figure 10.10. The function w = z 2 maps the interior of a circle onto the interior of a cardioid. Example 3. The function w = √ z maps the interior of a circle onto the interior of the right branch of a lemniscate (Fig. 10.11). The circle r =cosϕ is taken to the right branch of the lemniscate ρ = √ cos 2θ, where θ = 1 2 ϕ. X O 11 O U Y V Figure 10.11. The function w = √ z maps the interior of a circle onto the interior of the right branch of a lemniscate. Example 4. The function w =–ln(1 – z) maps the interior of the unit circle onto the interior of the curve u =–ln(2 cos v) (Fig. 10.12). Example 5. The function w =ln z – 1 1 + z maps the upper half-plane onto the strip 0 <Imz < π (Fig. 10.13). The function z =–coth 1 2 w specifies the inverse mapping of the strip 0 <Imz < π onto the upper half-plane. 422 FUNCTIONS OF COMPLEX VARIABLE X O 1ln π π 2 2 2 O U Y V Figure 10.12. The function w =–ln(1 – z) maps the interior of the unit circle onto the interior of the curve u =–ln(2 cos v). X πi O U Y V Figure 10.13. The function w =ln z – 1 1 + z maps the upper half-plane onto the strip 0 <Imz < π. 10.2.1-3. Linear-fractional mappings. The mappings given by linear-fractional functions w = az + b cz + d ,(10.2.1.2) where a, b, c,andd are complex constants and ad – bc ≠ 0, are called linear-fractional mappings. The function (10.2.1.2) is defined on the extended complex plane. (Its value at the point z =–d/c is definedtobe∞, and the value at the point z = ∞is defined to be a/c.) A linear-fractional function defines a schlicht mapping of the extended z-plane onto the extended w-plane. Linear-fractional functions are the only functions with this property. Points z and z ∗ are said to be symmetric about the circle C 0 : |z – z 0 | = R 0 if they lie on the same ray passing through z 0 and |z – z 0 ||z ∗ – z 0 | = R 2 0 . The transformation taking each point z to the point z ∗ symmetric to z about the circle C 0 is called the symmetry,ortheinversion, about the circle. Points z and z ∗ are symmetric about a circle C 0 if and only if they are the vertices of a pencil of circles orthogonal to the circle C 0 . T HEOREM. An arbitrary linear-fractional function w = az + b cz + d , ad – bc ≠ 0, defines a schlicht conformal mapping of the extended z -plane onto the extended w -plane. This mapping transforms any circle on the extended z -plane into a circle on the extended w -plane (the circular property) and transforms any pair of points symmetric about a circle C into a pair of points symmetric about the image of the circle C (preservation of symmetric points). . mapping of one of the domains onto the other. T HE MAIN THEOREM OF THE THEORY OF CONFORMAL MAPPINGS (RIEMANN THEOREM). For any simply connected domains D and D ∗ (with boundaries consisting of more. zeros and poles of f(z) . Moreover, the logarithmic residue of f(z) at a zero or a pole of f(z) is equal to the order of the zero or minus the order of the pole, respectively. The residue of a. conformal mapping preserves the angles between intersecting curves at points of intersection. T HEOREM. A function w = f(z) is a conformal mapping of a domain D if and only if it is analytic and