Handbook of mathematics for engineers and scienteists part 64 pps

10.1. BASIC NOTIONS 409 If f(z)andg(z) are analytic functions in a simply connected domain D and z = a and z = b are arbitrary points of the domain D, then the formula of integration by parts holds:  b a f(z) dg(z)=f (b)g(b)–f(a)g(a)–  b a g(z) df (z). (10.1.2.24) If an analytic function z = g(w) determines a single-valued mapping of a curve  C onto acurveC,then  C f(z) dz =   C f(g(w))g  w (w) dw.(10.1.2.25) C AUCHY’S THEOREM FOR A SIMPLY CONNECTED DOMAIN. If a function f(z) is analytic in a simply connected domain D bounded by a contour C and is continuous in D ,then  C f(z) dz = 0 . CAUCHY’S THEOREM FOR A MULTIPLY CONNECTED DOMAIN. If a function f(z) is analytic in a multiply connected domain D bounded by a contour Γ consisting of several closed curves and is continuous in D ,then  Γ f(z) dz = 0 provided that the sense of all curves forming Γ is chosen in such a way that the domain D lies to one side of the contour. If a function f(z) is analytic in an n-connected domain D and continuous in D,and C is the boundary of D, then for any interior point z of this domain the Cauchy integral formula holds: f(z)= 1 2πi  C f(ξ) ξ – z dξ.(10.1.2.26) (Here integration is in the positive sense of C; i.e., the domain D lies to the left of C.) Under the same assumptions as above, formula (10.1.2.26) implies expressions for the value of the derivative of arbitrary order of the function f (z) at any interior point z of the domain: f (n) z (z)= n! 2πi  C f(ξ) (ξ – z) n+1 dξ (n = 1, 2, ). (10.1.2.27) For an arbitrary smooth curve C, not necessarily closed, and for a function f(ξ)every- where continuous on C, possibly except for finitely many points at which this function has an integrable discontinuity, the right-hand side of formula (10.1.2.26) defines a Cauchy-type integral. The function F (z) determined by a Cauchy-type integral is analytic at any point that does not belong to C.IfC divides the plane into several domains, then the Cauchy-type integral generally determines different analytic functions in these domains. Formulas (10.1.2.26) and (10.1.2.27) allow one to calculate the integrals  C f(ξ) ξ – z dξ = 2πif(z),  C f(ξ) (ξ – z) n dξ = 2πi n! f (n) z (z). (10.1.2.28) Example 4. Let us calculate the integral  C Im zdz, where C is the semicircle |z| = 1, 0 ≤ argz ≤ π (Fig. 10.4). X 11 Y C Figure 10.4. The semicircle |z| = 1, 0 ≤ arg z ≤ π. 410 FUNCTIONS OF COMPLEX VARIABLE Using formula (10.1.2.21), we obtain  C Im zdz=  C y(dx + idy)=  C ydx+ i  C ydy=  –1 1 √ 1 – x 2 dx – i  –1 1 xdx=– π 2 . Example 5. Let us calculate the integral  C dz z – z 0 , where C is the circle of radius R centered at a point z 0 with anticlockwise sense. Using the Cauchy integral formula (10.1.2.28), we obtain  C 1 z – z 0 dz = 2πi. Example 6. Let us calculate the integral  C dz z 2 + 1 , where C is the circle of unit radius centered at the point i with anticlockwise sense. To apply the Cauchy integral formula (10.1.2.26), we transform the integrand as follows: 1 1 + z 2 = 1 (z – i)(z + i) = 1 z + i 1 z – i = f(z) z + i , f(z)= 1 z + i . The function f(z)=1/(z + i) is analytic in the interior of the domain under study and on its boundary; hence the Cauchy integral formula (10.1.2.26) and the first of formulas (10.1.2.28) hold. From the latter formula, we obtain  C dz z 2 + 1 =  C f(z) z – i dz = 2πif(i)=2πi 1 2i = π. Formulas (10.1.2.26) and (10.1.2.27) imply the Cauchy inequalities |f (n) z (z)| ≤ n! 2π     C f(ξ) (ξ – z) n+1 dξ    ≤ n!Ml 2πR n+1 ,(10.1.2.29) where M =max z D   f(z)   is the maximum modulus of the function f (z) in the domain D, R is the distance from the point z to the boundary C,andl is the length of the boundary C. If, in particular, f(z) is analytic in the disk D = |z – z 0 | < R, and bounded in ¯ D,thenwe obtain the inequality |f (n) z (z 0 )| ≤ n!M R n (n = 0, 1, 2, ). (10.1.2.30) M ORERA’S THEOREM. If a function f(z) is continuous in a simply connected domain D and  C f(z) dz = 0 for any closed curve C lying in D ,then f(z) is analytic in the domain D. 10.1.2-5. Taylor and Laurent series. If a series ∞  n=0 f n (z)(10.1.2.31) of analytic functions in a simply connected domain D converges uniformly in this domain, then its sum is analytic in the domain D. If a series (10.1.2.31) of functions analytic in a domain D and continuous in D converges uniformly in D, then it can be differentiated termwise any number of times and can be integrated termwise over any piecewise smooth curve C lying in D. 10.1. BASIC NOTIONS 411 A BEL’S THEOREM. If the power series ∞  n=0 c n (z – a) n (10.1.2.32) converges at a point z 0 , then it also converges at any point z satisfying the condition |z – a| < |z 0 – a| . Moreover, the series converges uniformly in any disk |z – a| < q|z 0 – a| , where 0 < q < 1 . It follows from Abel’s theorem that the domain of convergence of a power series is an open disk centered at the point a; moreover, this disk can fill the entire plane. The radius of this disk is called the radius of convergence of a power series. The sum of the power series inside the disk of convergence is an analytic function. Remark. The radius of convergence R can be found by the Cauchy–Hadamard formula 1 R = lim n→∞ n  |c n |, where lim denotes the upper limit. If a function f(z) is analytic in the open disk D of radius R centered at a point z = a, then this function can be represented in this disk by its Taylor series f(z)= ∞  n=0 c n (z – a) n , whose coefficients are determined by the formulas c n = f (n) z (a) n! = 1 2πi  C f(ξ) (ξ – z) n+1 dξ (n = 0, 1, 2, ), (10.1.2.33) where C is the circle |z – a| = qR, 0 < q < 1. In any closed domain belonging to the disk D, the Taylor series converges uniformly. Any power series expansion of an analytic function is its Taylor expansion. The Taylor series expansions of the functions given in Paragraph 10.1.2-3 in powers of z have the form e z = 1 + z + z 2 2! + z 3 3! + (|z| < ∞), (10.1.2.34) cos z = 1 – z 2 2! + z 4 4! – ,sinz = z – z 3 3! + z 5 5! – (|z| < ∞), (10.1.2.35) cosh z = 1 + z 2 2! + z 4 4! + ,sinhz = z + z 3 3! + z 5 5! + (|z| < ∞), (10.1.2.36) ln(1 + z)=z – z 2 2! + z 3 3! – (|z| < 1), (10.1.2.37) (1 + z) a = 1 + az + a(a – 1) 2! z 2 + a(a – 1)(a – 2) 3! z 3 + (|z| < 1). (10.1.2.38) The last two expansions are valid for the single-valued branches for which the values of the functions for z = 0 are equal to 0 and 1, respectively. 412 FUNCTIONS OF COMPLEX VARIABLE Remark. Series expansions (10.1.2.34)–(10.1.2.38) coincide with analogous expansions of the corre- sponding elementary functions of the real variable (see Paragraph 8.3.2-3). To obtain the Taylor series for other branches of the multi-valued function Ln(1 + z), one has to add the numbers 2kπi, k = 1, 2, to the expression in the right-hand side: Ln(1 + z)=z – z 2 2! + z 3 3! – + 2kπ. The domain of convergence of the function series ∞  n=–∞ c n (z – a) n is a circular annulus K : r < |z – a| < R,where0 ≤ r ≤ ∞ and 0 ≤ R ≤ ∞. The sum of the series is an analytic function in the annulus of convergence. Conversely, in any annulus K where the function f(z) is analytic, this function can be represented by the Laurent series expansion f(z)= ∞  n=–∞ c n (z – a) n with coefficients determined by the formulas c n = 1 2πi  γ f(ξ) (ξ – z) n+1 dξ (n = 0, 1, 2, ), (10.1.2.39) where γ is the circle |z –a| = ρ, r < ρ < R. In any closed domain contained in the annulus K, the Laurent series converges uniformly. The part of the Laurent series with negative numbers, –1  n=–∞ c n (z – a) n = ∞  n=1 c –n (z – a) n ,(10.1.2.40) is called its principal part, and the part with nonnegative numbers, ∞  n=0 c n (z – a) n ,(10.1.2.41) is called the regular part. Any expansion of an analytic function in positive and negative powers of z – a is its Laurent expansion. Example 7. Let us consider Laurent series expansions of the function f(z)= 1 z(1 – z) in a Laurent series in the domain 0 < |z| < 1. This function is analytic in the annulus 0 < |z| < 1 and hence can be expanded in the corresponding Laurent series. We write this function as the sum of elementary fractions: f(z)= 1 z(1 – z) = 1 z + 1 1 – z . Since |z| < 1, we can use formula (10.1.2.39) and obtain the expansion 1 z(1 – z) = 1 z + 1 + z + z 2 + Example 8. Let us consider Laurent series expansions of the function f(z)=e 1/z in a Laurent series in a neighborhood of the point z 0 = 0. To this end, we use the well-known expan- sion (10.1.2.34), where we should replace z by 1/z. Thus we obtain e 1/z = 1 + 1 1!z + 1 2!z 2 + + 1 n!z n + (z ≠ 0). 10.1. BASIC NOTIONS 413 10.1.2-6. Zeros and isolated singularities of analytic functions. A point z = a is called a zero of a function f (z)iff(a)=0.Iff (z) is analytic at the point a and is not zero identically, then the least order of nonzero coefficients in the Taylor expansion of f(z) centered at a, in other words, the number n of the first nonzero derivative f (n) (a), is called the order of zero of this function. In a neighborhood of a zero a of order n, the Taylor expansion of f (z)hastheform f(z)=c n (z – a) n + c n+1 (z – a) n+1 + (c n ≠ 0, n ≥ 1). In this case, f(z)=c n (z – a) n g(z), where the function g(z) is analytic at the point a and g(a) ≠ 0.Afirst-orderzeroissaidtobesimple. The point z = ∞ is a zero of order n for a function f(z)ifz = 0 is a zero of order n for F (z)=f(1/z). If a function f(z) is analytic at a point a and is not identically zero in any neighborhood of a, then there exists a neighborhood of a in which f(z) does not have any zeros other than a. U NIQUENESS THEOREM. If functions f(z) and g(z) are analytic in a domain D and their values coincide on some sequence a k of points converging to an interior point a of the domain D ,then f(z) ≡ g(z) everywhere in D . ROUCH ´ E’S THEOREM. If functions f(z) and g(z) are analytic in a simply connected domain D bounded by a curve C , are continuous in D , and satisfy the inequality |f(z)| > |g(z)| on C , then the functions f(z) and f(z)+g(z) have the same number of zeros in D . A point a is called an isolated singularity of a single-valued analytic function f(z)if there exists a neighborhood of this point in which f(z) is analytic everywhere except for the point a itself. The point a is called 1. A removable singularity if lim z→a f(z) exists and is finite. 2. A pole if lim z→a f(z)=∞. 3. An essential singularity if lim z→a f(z) does not exist. A necessary and sufficient condition for a point a to be a removable singularity of a function f(z) is that the Laurent expansion of f(z) around a does not contain the principal part. If a function f(z) is bounded in a neighborhood of an isolated singularity a,thena is a removable singularity of this function. A necessary and sufficient condition for a point a to be a pole of a function f (z)isthat the principal part of the Laurent expansion of f(z) around a contains finitely many terms: f(z)= c –n (z – a) n + + c –1 (z – a) + ∞  k=0 c k (z – a) k .(10.1.2.42) The order of a pole a of a function f(z)isdefined to be the order of the zero of the function F (z)=f (1/z). If c –n ≠ 0 in expansion (10.1.2.42), then the order of the pole a of the function f (z) is equal to n.Forn = 1,wehaveasimple pole. A necessary and sufficient condition for a point a to be an essential singularity of a function f(z) is that the principal part of the Laurent expansion of f(z) around a contains infinitely many nonzero terms. S OKHOTSKII’S THEOREM. If a is an essential singularity of a function f(z) , then for each complex number A there exists a sequence of points z k → a such that f(z k ) → A . 414 FUNCTIONS OF COMPLEX VARIABLE Example 9. Let us consider some functions with different singular points. 1 ◦ . The function f(z)=(1 –cosz)/z 2 has a removable singularity at the origin, since its Laurent expansion about the origin, 1 –cosz z 2 = 1 2 – z 2 24 + z 4 720 – , does not contain the principal part. 2 ◦ . The function f(z)=1/(1+e z 2 )hasinfinitely many poles at the points z = √ (2k + 1)πi (k = 0, 1, 2, ). All these poles are simple poles, since the function 1/f(z)=1 + e z 2 has simple zeros at these points. (Its derivative is nonzero at these points.) 3 ◦ . The function f(z)=sin(1/z) has an essential singularity at the origin, since the principal part of its Laurent expansion sin 1 z = 1 z – 1 z 3 3! + contains infinitely many terms. The following two simplest classes of single-valued analytic functions are distinguished according to the character of singular points. 1. Entire functions. A function f(z)issaidtobeentire if it does not have singular points in the finite part of the plane. An entire function can be represented by an everywhere convergent power series f(z)= ∞  n=0 c n z n . An entire function can have only one singular point at z = ∞. If this singularity is a pole of order n,thenf(z) is a polynomial of degree n.Ifz = ∞ is an essential singularity, then f (z) is called an entire transcendental function.Ifz = ∞ is a regular point (i.e., f (z) is analytic for all z), then f (z) is constant (Liouville’s theorem). All polynomials, the exponential function, sin z,cosz, etc. are examples of entire functions. Sums, differences, and products of entire functions are themselves entire functions. 2. Meromorphic functions. A function f (z)issaidtobemeromorphic if it does not have any singularities except for poles. The number of these poles in each finite closed domain D is always finite. Suppose that a function f(z) is analytic in a neighborhood of the point at infinity. The definition of singular points can be generalized to this function without any changes. But the criteria for the type of a singular point at infinity related to the Laurent expansion are different. T HEOREM. In the case of a removable singularity at the point at infinity, the Laurent expansion of a function f(z) in a neighborhood of this point does not contain positive powers of z . In the case of a pole, it contains fi nitely many positive powers of z . In the case of an essential singularity, it contains infinitely many powers of z . Let f(z) be a multi-valued function defined in a neighborhood D of a point z = a except possibly for the point a itself, and let f 1 (z), f 2 (z) be its branches, which are single-valued continuous functions in the domain where they are defined. The point a is called a branch point (ramification point) of the function f (z)iff(z) passes from one branch to another as the point z goes along a closed curve around the point z in a neighborhood of D.If the original branch is reached again for the first time after going around this curve m times (in the same sense), then the number m – 1 is called the order of the branch point, and the point a itself is called a branch point of order m – 1. 10.1. BASIC NOTIONS 415 If all branches f k (z)tendtothesamefinite or infinite limit as z → a, then the point a is called an algebraic branch point. (For example, the point z = 0 is an algebraic branch point of the function f(z)= m √ z.) In this case, the single-valued function F (z)=f(z m + a) has a regular point or a pole for z = 0. If the limit of f k (z)asz → a does not exist, then the point a is called a transcendental branch point. For example, the point z = 0 is a transcendental branch point of the function f(z)=exp( m  1/z ). In a neighborhood of a branch point a of finite order, the function f(z) can be expanded in a fractional power series (Puiseux series) f(z)= ∞  k=–∞ c k (z – a) k/m .(10.1.2.43) If a new branch is obtained each time after going around this curve (in the same sense), then the point a is called a branch point of infinite order (a logarithmic branch point). For example, the points z = 0 and z = ∞ are logarithmic branch points of the multivalued function w =Lnz. A logarithmic branch point is classified as a transcendental branch point. For a ≠ ∞, the expansion (10.1.2.43) contains finitely many terms with negative k (infinitely many in the case of a transcendental point). 10.1.2-7. Residues. The residue res f(a) of a function f(z) at an isolated singularity a is defined as the number res f(a)= 1 2πi  C f(z) dz,(10.1.2.44) where the integral is taken in the positive sense over a contour C surrounding the point a and containing no other singularities of f(z) in the interior. Remark. Residues are sometimes denoted by res[f(z); a]orres z=a f(z). The residue res f(a)ofafunctionf (z) at a singularity a is equal to the coefficient of (z – a) –1 in the Laurent expansion of f (z) in a neighborhood of the point a, res f(a)= 1 2πi  C f(z) dz = c –1 .(10.1.2.45) Basic rules for finding the residues: 1. The residue of a function at a removable singularity is always zero. 2. If a is a pole of order n,then res f(a)= 1 (n – 1)! lim z→a d n–1 dz n–1  f(z)(z – a) n  .(10.1.2.46) 3. For a simple pole (n = 1), res f(a) = lim z→a  f(z)(z – a) n  .(10.1.2.47) . NOTIONS 409 If f(z)andg(z) are analytic functions in a simply connected domain D and z = a and z = b are arbitrary points of the domain D, then the formula of integration by parts holds:  b a f(z). positive sense of C; i.e., the domain D lies to the left of C.) Under the same assumptions as above, formula (10.1.2.26) implies expressions for the value of the derivative of arbitrary order of the. analytic in the interior of the domain under study and on its boundary; hence the Cauchy integral formula (10.1.2.26) and the first of formulas (10.1.2.28) hold. From the latter formula, we obtain  C dz z 2 +