Functions of a Complex Variable and Integral Transforms

Functions of a Complex Variable and Integral Transforms Department of Mathematics Zhou Lingjun

Textbook Functions of Complex Analysis with Applications to Engineering and Science, 3rd Edition. A. D. Snider & E. B. Saff. References Complex Analysis, Third Edition, Lars V.Ahlfors

What is the course about? The main roles of this course are single variable functions which are defined on a set of complex numbers. We concentrate on a group of beautiful functions, which are called analytic functions. Using the property of analytic functions, many difficult problems can be solved. The integral transforms are based on the single complex variable functions, which are powerful tools to solve differential equations.

The History of Numbers Natural numbers are created to count. For fixed natural numbers m > n, the equation m + x = n will not have a solution in natural numbers. In order to solve such equations, the negative integers are introduced. For fixed integer n and nonzero integer m, the equation mx = n may not have a solution in integer numbers. In order to solve such equations, the rational numbers are introduced. For fixed positive rational number q, the equation x 2 = q may not have a solution in the rational numbers. In order to solve such equations, the real numbers are introduced. For fixed negative real number q, the equation x 2 = q will not have a solution in the real numbers. In order to solve such equations, the complex number are introduced.

Chapter 1 Complex Numbers

1.1 The Algebra of Complex Numbers

Introduction to Complex Numbers We call z = x + iy a complex number, where x, y are real numbers and i is a symbol of character. The real number x, y are called the real and imaginary parts of z, which are denoted as Re z and Im z respectively. Two complex numbers are equal iff Re z 1 = Re z 2 and Im z 1 = Im z 2. The set of all the complex numbers is denoted as C. We use the real number x to denote x + i0 in brief, in which sense the set of real numbers R can be regarded as a subset of the set C. We use iy to denote 0+iy in brief, which is called a pure imaginary number.

Fundamental Operations Addition and Multiplication: For z 1 = x 1 + iy 1, z 2 = x 2 + iy 2, we define z 1 + z 2 = (x 1 + x 2 ) + i(y 1 + y 2 ), z 1 z 2 = (x 1 x 2 y 1 y 2 ) + i(x 1 y 2 + x 2 y 1 ). The definition is natural, since i is treated as a symbol and i 2 is replaced by 1. For z = x+iy, denote z = ( x)+i( y) = ( 1)z, then z+( z) = 0. For z = x + iy 0, denote z 1 = x iy x 2 +y 2, then zz 1 = 1.

Subtraction and Division: For z 1, z 2 C, we define z 1 z 2 = z 1 + ( z 2 ). Moreover if z 2 0, we define z 1 z2 = z 1 z2 1. Operational Rules Commutative Law of Addition z 1 + z 2 = z 2 + z 1 Associative Law of Addition (z 1 + z 2 ) + z 3 = z 1 + (z 2 + z 3 ) Commutative Law of Multiplication z 1 z 2 = z 2 z 1 Associative Law of Multiplication (z 1 z 2 )z 3 = z 1 (z 2 z 3 ) Distributive Law (z 1 + z 2 )z 3 = z 1 z 3 + z 2 z 3 The complex number set C with the addition and multiplication defined above is a field.

1.2 Point Representation of Complex Numbers Each complex number z = x + iy can be associated a point (x, y) in the plane. For z = x + iy, we define z = x 2 + y 2, which is called the modulus (or absolute value or norm) of z. For x = x+iy, the complex conjugate of z is defined by z = x iy. z = z. z is real if and only if z = z.

Vectors and Polar Forms Vectors and Complex numbers: A complex number z = x + iy can be regarded as a vector. z can be expressed in Cartesian coordinate (x, y), or in terms of the polar coordinate (r, θ)

r = z. θ is determined up to an integer multiple of 2π. We call the value of any of these angles an argument of z, denoted Arg z. The argument lies in the interval ( π, π) is known as the principle value of the argument, denoted arg z. Polar Form: z = r(cos θ + sin θ). Triangle Inequality: z 1 z 2 z 1 + z 2 z 1 + z 2. z 1 z 2 = z 1 z 2, z 1 z2 = z 1 z 2. Arg (z 1 + z 2 ) = Arg z 1 + Arg z 2.

z = z, arg z = arg z. Re z z, Im z z. z 1 w 1 + + z n w n z 1 2 + + z n 2 w 1 2 + + w n 2.

Powers and Roots For z and a positive integer n, we define z n = z z. }{{} For nonzero z and a positive integer n, we define z n = (z n ) 1. De Moivre Formula. If z = r(cos θ + sin θ) and n is an integer, then z n = r n (cos nθ + sin nθ). For nonzero z = r(cos θ + sin θ) and a positive integer nthe nth roots of z are given by the n complex numbers z k = n [ ) )] r cos + i sin, k = 0, 1,..., n 1. ( θ n + 2kπ n ( θ n + 2kπ n The roots form the vertices of a regular polygon. n

1.3 Planar Sets We call the set N δ (z 0 ) = {z z z 0 < δ} an open disk of z 0. A point z 0 is called an interior point of the set S, if there exists an open disk of z 0 which is contained in S. A set S is called an open set, if each point of S is an interior point of S. A point z 0 is called an exterior point of the set S, if z 0 is the interior point of S c (the complement set of S). A point z 0 is called a boundary point of the set S, if z 0 is neither an interior point nor an exterior point. The set of all boundary points of E is called the boundary of S, which is denoted by S. Denote S = S S, which is called the closure of S.

A open set D is called a domain (or region) if it is connected, which means each pair z 1, z 2 D can be joined by a polygonal path lying in D. In our lectures, we always use D to denote a domain without further interpretation. Theorem. Suppose u(x, y) is a real-valued function defined in a domain D. If the first partial derivatives of u satisfy u x = u y = 0 at all point of D, then u is a constant in D.

Contours A curve γ in domain D is the range of a continuous mapping z(t) from [a, b] to D. z(t) is called a parametrization of γ. A curve γ is said to be a smooth arc if it is the range of some continuous complex-valued function z = z(t), (a t b) that satisfies the following conditions: 1. z(t) has a continuous derivative on [a, b], 2. z (t) never vanishes on [a, b], 3. z(t) is one-to-one on [a, b]. A curve γ is called a smooth closed curve if it is the range of some continuous complex-valued function z = z(t), (a t b), satisfying conditions (1) and (2) and the following:

3. z(t) is one-to-one on [a, b), but z(b) = z(a) and z (b) = z (a). The phrase smooth curve means either a smooth arc or a smooth closed curve. A smooth arc is called a directed smooth arc if one of its endpoints is specified as the initial point. A smooth closed curve whose points have been ordered is called a directed smooth closed curve. The phrase directed smooth curve means either a directed smooth arc or a directed smooth closed curve. A contour Γ is either a single point z 0 or a finite sequence of directed smooth curve (γ 1, γ 2,..., γ n ) such that the terminal point of γ k coincides with the initial point of γ k+1 for each k = 1, 2,..., n 1. In this case one can write Γ = γ 1 + γ 2 + + γ n.

Γ is said to be a closed contour or a loop if its initial and terminal points coincide. A simple closed contour is a closed contour with no multiple points other than its initial-terminal point. Jordan Curve Theorem. Any simple closed contour divides the plane into exactly two domains, each having the curve as its boundary. One of these domains, called the interior, is bounded and the other, called exterior is unbounded. The direction along a simple closed contour Γ can be completely specified by declaring its initial-terminal point and stating which domain lies to the left of an observer tracing out the points in order. When the interior domain lies to the left, we say that Γ is positively oriented. Otherwise Γ is said to be negatively oriented. A positively orientation generalizes the concept of counterclockwise motion.

Riemann Sphere and Stereographic Projection For each point A in the equatorial plane, we construct the line

passing through the the north pole N = (0, 0, 1) of the unit sphere and A. This line pierces the unit sphere in exactly one point P (A). For each point on the sphere other than the north pole N, we construct the line passing through these two points. This line pierces the equatorial plane in exactly one point. Therefore the map P, called stereographic projection, is a bijection from the equatorial plane to the unit sphere without the north pole. If we identify the equatorial plane as the complex plane, the unit sphere is call the Riemann sphere. N does not arise as the projection of any point of in the complex plane. However we find P 1 (z) tends to N as z. We set is the image of N under the stereographic projection, and the complex plane together with is called the extended complex plane, which is denoted by C.

1.4 Functions of a Complex Variable In this lecture, we usually consider a function w = f(z) defined on a subset of the complex plane C, and its range also lies in C. Let z = x + iy (x, y R), then f(z) can be written as f(z) = u(x, y) + iv(x, y), where u(x, y), v(x, y) are real-valued functions. Unlike the real variable function, we cannot draw a graph of a complex variable function, since both variable z and value w are located in a plane. We usually draw the z and w plane separately.

Chapter 2 Analytic Functions

2.1 Limits and continuity The definitions of limits and continuity in the sense of complex variable functions are the same as the relevant definitions in calculus. The properties of limits are also the same as the relevant properties in calculus. The function f(z) is continuous iff both u(x, y) and v(x, y), regarding as real variable functions, are continuous. A continuous complex variable function do not have more special properties than a continuous real variable function.

2.2 Analyticity Let f be a complex-valued function defined in a neighborhood of z 0. Then the derivative of f at z 0 is given by f (z 0 ) = lim z z0 f(z) f(z 0 ) z z 0, provided this limit exists. Such an f is said to be differentiable at z 0. The properties of derivatives are almost the same as the relevant properties in calculus, for example the chain rule. A function f is called analytic in a domain D C if it has a derivative at each point in D. If f(z) is analytic on the whole complex plane, then it is said to be entire.

The Cauchy-Riemann Equations Theorem 1. The necessary condition for a function f(z) = u(x, y)+ iv(x, y) to be differentiable at z 0 is that the Cauchy-Riemann e- quations hold at z 0. u x = v y, u y = v x Theorem 2. Let f be a complex-valued function defined in some open set G containing z 0. If the first partial derivatives of u and v exist in G, and satisfy the the Cauchy-Riemann equations at z 0, then f is differentiable at z 0. (The proof is highly involved.) Corollary. f is analytic in some region D if and only if u, v satisfy the Cauchy-Riemann equations in D.

2.4 The Elementary Functions 1. Exponential function. e z is an entire function, and (e z ) = e z. e z is periodic function, and the period is 2πi. e z 1e z 2 = e z 1+z 2. e z is one-to-one on each strip {x + iy : a < y < a + h, h < 2π}.

2. The Trigonometric Functions For z C, we define cos z = eiz + e iz, sin z = eiz e iz. 2 2i Most formulae for cos z and sin z hold in C. sin z and cos z are periodic functions, and the period is 2π. sin z and cos z are entire function, and (sin z) = cos z, (cos z) = sin z. sin z and cos z are boundless, that is to say that sin z 1, cos z 1 do not hold in C.

3. The Hyperbolic Functions For any z C, we define cosh z = ez + e z, sinh z = ez e z. 2 2 cosh 2 z sinh 2 z = 1. sinh z and cosh z are periodic functions, and the period is 2πi. sin z and cos z are entire function, and (sinh z) = cosh z, (cosh z) = sinh z. sin iz = i sinh z, sinh iz = i sin z, cos iz = cosh z, cosh iz = cos z.

4. The Logarithmic Function For any nonzeoro z C, we define Log z = log z + iarg z, log z = log z + i arg z. Log z is a multiple valued function, and it solves the equation e w = z with respect of w. log z is called the principal value of Log z. log z is analytic in C \ {Re z 0.}, and (log z) = 1 z. log z is not continuous in the set {Re z < 0.} Remark The formulae for log z for real variable do not always hold for complex variable. We should treat it very carefully.

5. Complex Powers For any nonzero z C and α C, we define z α = e αlog z. z α is usually a multiple valued function. When α is an integer, z α is single-valued and the definition above consists with the definition in Section 1.4. When α = p q (p Z, q N) is rational, z p q values, which are q-th root of z p. takes exactly q distinct

When α is irrational, z α takes infinity many values. In order to make z α be single-valued, we need to restrict Log z on a single-valued branch. For example we define the principal value of z α to be e α log z, then the principal value is analytic in C \ {Re z 0.}, and its derivative is αz α 1.

6. Inverse Trigonometric Functions The inverse trigonometric function can be interpreted as logs. Arcsin z = ilog [iz + (1 z 2 ) 1 2], Arccos z = ilog [z + (z 2 1) 1 2], Arctan z = i 2 Log i + z i z. We can obtain the principal value by choosing the principal values of both the square root and the logarithm.

2.5 Cauchy s Integral Theorem Definition Let f(z) = u(x, y) + iv(x, y) be a complex-valued function defined on the directed smooth curve γ, then we define complex integral of f along C as the following: γ f(z) dz = γ u dx v dy + i γ v dx + u dy. If γ is parameterized as z = z(t) (a t b), then the integral can be computed as the following: γ f(z) dz = b a f(z(t))z (t) dt.

Max-Length inequality: If f is continuous on the contour Γ, then γ f(z) dz γ f(z) ds length(γ) max γ f. Newton-Leibniz Axiom. Suppose that the function f(z) has an antiderivative F (z) throughout a domain D, i.e. F (z) = f(z) for each z D. Then for any contour Γ lying in D, with initial point z 1 and terminal point z 2, we have f(z) dz = F (z 2 ) F (z 1 ). γ Theorem. Let f be defined in a domain D. The following are equivalent: (1) f has an antiderivative in D.

(2) Every loop integral of f in D vanishes. (3) The contour integrals of f are independent of path in D. Indefinite Integral. Suppose that f is continuous in a domain D and the contour integrals of f are independent of path in D. Let F (z) be the integral of f along some contour Γ in D joining z 0 and z. Then F (z) is an antiderivative of f(z).

Cauchy s Integral Theorem In this section, we always assume that D is a bounded domain and its boundary D is the union of finite closed contours. The external boundary is always taken the positive orientation, while the internal boundary is taken the negative oriented, so

that D always lies to the left of an observer tracing along the boundary. Cauchy s Integral Thoerem. Suppose f is analytic in D and continuous up to and including its boundary, then f(z) dz = 0. D Proof. By using Green s theorem, one can find Ω f(z) dz = Ω u dx v dy + i = Ω ( u y + v ) x dxdy + i Ω Ω ( u x v y u dy + v dx ) dxdy. Therefore the result is led by the Cauchy-Riemann equations. Remark. In order to use Green s Theorem, u and v are required to have continuous partial derivatives. Actually it can be proved but the proof is quite involved.

A simply connected domain D is a domain having the following property: If Γ is any simple closed contour lying in D, then the domain interior to Γ lies wholly in D. Roughly speaking, a simply connected domain has no hole inside. Corollary 1. Suppose that f is analytic in a simply connected domain D and Γ is any loop in D, then f(z)dz = 0. Γ Corollary 2. In a simply connected domain, an analytic function has an antiderivative, its contour integrals are independent of path, and its loop integrals vanish.

2.6 Cauchy s Integral Formula and Its Consequences Cauchy s Integral Formula. Suppose that z 0 lies in an open domain D and f(z) is analytic in D and continuous up to and including its boundary, then f(z 0 ) = 1 2πi D f(z) z z 0 dz.

Proof. Let N ε (z 0 ) be a neighborhood lying in D. Using Cauchy integral theorem with the domain D\N ε (z 0 ) and the function f(z) z z, 0 one can see f(z) dz = 0. z z 0 (D\N ε (z 0 ))

On the other hand, since therefore (D\N ε (z 0 )) f(z) z z 0 dz = D D f(z) z z 0 dz = f(z) z z 0 dz N ε (z 0 ) N ε (z 0 ) f(z) z z 0 dz. f(z) z z 0 dz, Set z(θ) = z 0 + εe iθ (0 θ 2π), then N ε (z 0 ) f(z) z z 0 dz = 2π 0 f(εe iθ ) εe iθ εe iθ i dθ = Set ε 0, the continuity of f implies D f(z) z z 0 dz = 2πif(z 0 ). 2π 0 f(z 0 + εe iθ ) dθ.

Higher Derivatives. Suppose that z 0 lies in a open domain D and f(z) is analytic in D and continuous up to and including its boundary, then f (n) (z 0 ) = n! 2πi D f(z) (z z 0 ) n+1 dz. Proof. It is led by differentiating both sides of Cauchy integral formula with respect of z 0. Morera s Theorem. If f is continuous in a domain D and if each loop integral of f in D vanishes, then f is analytic in D.

2.7 Taylor Series Geometric Series. { z < 1}. The power series n=0 z n converges to 1 1 z if Comparison Test. Suppose that the terms c j satisfy the inequality c j M j for all integer j larger than some number J. Then if the series j=0 M j converges, so does j=0 c j. Ratio Test. Suppose that the terms of the series j=0 c j have the property that the ratios c j+1 /c j approach a limit L as j. Then the series converges if L < 1 and diverges if L > 1. Definition. The sequence {F n (z)} n=1 is said to converge uniformly to F (z) on the set T if for any ε > 0 there exists an integer N such that when n > N, F (z) F n (z) < ε for all z in T.

A power series is a series of the form and z 0 are fixed complex numbers. n=0 a n (z z 0 ) n, where a n For any power series n=0 a n (z z 0 ) n, there exists R > 0 (may be infinity), called the radius of convergence, such that (1)the series converges in { z z 0 < R}, and converges uniformly in any closed subset of the disk. (2)the series diverges in z z 0 > R. The convergence radius can be computed by the formula ) ( ) 1 1 n a R = a n = n+1, ( lim n if the limit exist. In general, R = ( lim n lim n n a n ) 1. a n

The convergence is uniform and absolute in every closed disc in D, that is to say the series can be differentiated and integrated termwise in D. A power series a n (z z 0 ) n is analytic in the convergence domain. n=0 If f is analytic in D r = { z z 0 < r}, then for all z D r, f can be uniquely expressed as the following series: f(z) = n=0 a n (z z 0 ) n, where a n = f (n) (z 0 ) n!. Such series is called Taylor series of f around z 0. Furthermore, the convergence of the series is uniform in any closed subset of D r.

Proof. For fixed positive ρ < r, by Cauchy integral formula, f(z) = 1 2πi w z 0 =ρ f(w) w z dw = 1 2πi w z 0 =ρ f(w) (w z 0 )(1 z z 0 w z ) dw. 0 Noting that z z 0 w z < 1, using the lemma above, integrating term 0 by term and using the formula for higher derivatives, we have f(z) = 1 ( ) f(w) n z z0 dw 2πi w z 0 w z 0 = = n=0 n=0 w z 0 =ρ 1 2πi w z 0 =ρ n=0 f (n) (z 0 ) (z z 0 ) n. n! f(w) (w z 0 ) dw n+1 (z z 0) n The uniqueness is following by differentiating both sides and valuing at z 0.

Remark. The theorem implies that the Taylor series will converge to f(z) everywhere inside the largest open disk, centered at z 0, over which f is analytic. Taylor series of some elementary functions. e z = cos z = sin z = z n n=0 n! = 1 + z + z2 2! + z3 3! +, z C, n=0 n=0 log(1 z) = ( 1) n (2n)! z2n = 1 z2 2! + z4 4! +, z C, ( 1) n+1 (2n + 1)! z2n+1 = z z3 3! + z5 5! +, z C, n=1 z n n = z z2 2 z3 3, z < 1. Again this is a remarkable property which is not shared by functions of real variables. A function need not have a convergent

Taylor series expansion even if it is infinitely differentiable. example, f(x) = e 1 x 2, x > 0, 0, x 0. For Theorem. If f is analytic at z 0, the Taylor series for f around z 0 can be obtained by termwise differentiation of the Taylor series for f around z 0 and converges in the same disk as the series for f. Historically, we call a function f(z) is analytic at z 0 if f(z) can be locally expressed as a power series f(z) = n=0 a n (z z 0 ) n around z 0. We now see analyticity is equivalent to differentiability.

Chapter 3 Isolated Singularities

3.1 Zeros and Singularities Zeros. A point z 0 is called a zero of order m for the analytic function f if f and its first m 1 derivatives vanish at z 0, but f (m) (z 0 ) 0. Assume that z 0 is a zero of order m of the analytic function f(z), then the Taylor series for f around z 0 takes the form f(z) = n=k a n (z z 0 ) n = (z z 0 ) k which implies the following proposition. n=0 a n+k (z z 0 ) n, Proposition. If z 0 is a zero for the analytic function f(z), then either f is identically zero in a neighborhood of z 0 or there is a punctured disk about z 0 in which f has no zeros.

By using the connectivity of D, we can obtain a stronger theorem. Theorem. If f is analytic in a domain D and z 0 is a zero for f, then either f is identically zero in D or there is a punctured disk about z 0 in which f has no zeros. That is to say the zeros for a nonconstant analytic function must be isolated. Corollary If f(z) and g(z) are analytic in D and f(z) = g(z) on a set which has an accumulation point in D, then f(z) identically equals to g(z). Singularity. A point is called a singularity of f(z), if f(z) fails to be analytic at this point. A singularity z 0 of f(z) is called isolated, if f(z) is analytic in some annulus {0 < z z 0 < ε}.

Example 1 0 is a isolated singularity of sin z z, 1 z or sin 1 z. Example 2 0 is a singularity of log z, but not isolated.

3.2 Laurent Series If f is analytic in the annulus D = {r < z z 0 < R}, then for each z D f(z) = + n= where the coefficients are given by c n = 1 2πi γ c n (z z 0 ) n, f(w) (w z 0 ) n+1dz and γ is any simply closed curve lying in D containing z 0 in its interior. This series is called Laurent series of f in the annulus D. Moreover such a series expansion is unique. Proof. For any z D, take ε > 0 such that z Ω = {r ε < z z 0 < R ε}. Since f is analytic in Ω and continuous in Ω,

using Cauchy integral formula, we find f(z) = 1 2πi Ω f(w) w z dw = 1 2πi w z 0 =R+ε f(w) w z dw 1 2πi w z 0 =r ε f(w) w z dw.

Noting that the following series hold for z Ω 1 w z = 1 w z 0 1 w z = 1 z z 0 1 1 z z = 0 w z 0 1 1 w z = 0 z z 0 n=0 n=1 the integral above can be rewritten as w z 0 =R+ε w z 0 =r ε f(w) w z dw = f(w) w z dw = n=0 n=1 (z z 0 ) n (w z 0 ) n+1, for w z 0 = R ε, (z z 0 ) n (w z 0 ) n 1 (z z 0 ) n, for w z 0 = r + ε, w z 0 =R+ε (z z 0 ) n w z 0 =r ε f(w) (w z 0 ) n+1 dw, f(w)(w z 0 ) n 1 dw.

Set one can see c n = 1 2πi w z 0 =R+ε 1 2πi w z 0 =r ε f(z) = f(w) dw, for n 0, (w z 0 ) n+1 f(w) dw, for n < 0, (w z 0 ) n+1 + n= c n (z z 0 ) n. Furthermore for any simple closed curve γ Ω, by Cauchy integral theorem, we have f(w) (w z 0 ) n+1 dw = f(w) (w z 0 ) n+1 dw = f(w) (w z 0 ) w z 0 =R+ε w z 0 =r ε γ n+1 dw. The uniqueness follows by dividing both sides of series by (z

z 0 ) n+1 and integrating along γ. Remark 1 The Laurent series of f depends on not only the center point z 0 but the annulus D, while the Taylor series only depends on the center point. Remark 2 It is usually very hard to compute the coefficients in Laurent series by integral. However we can use the formulae for Taylor series to get the coefficients in Laurent series, and it also gives us another way to compute the integrals. One of the significant results is residue theorem, which will be shown in the next chapter.

3.3 Three Types of Isolated Singularities Definition. If z 0 is an isolated singularity of f(z), then f(z) has a Laurent expansion as f(z) = + n= c n (z z 0 ) k in a punctured disc {0 < z z 0 < ε}. singularities into three categories. We classify the isolated (1) If a j = 0 for all j < 0, we say that z 0 is a removable singularity. (2) If a m 0 for some positive integer m but a j = 0 for all j < m, we say that z 0 is a pole of order m. (3) If a j 0 for an infinite number of negative values of j, we say that z 0 is an essential singularity of f.

Behavior near the isolated singularity. Let z 0 be an isolated singularity of f(z). (1) z 0 is removable iff lim z z0 f(z) exists finite. (2) z 0 is a pole iff lim z z0 f(z) =. (3) z 0 is an essential singularity iff lim z z0 f(z) does not exist. Theorem Let z 0 be an isolated singularity of f(z). (1) z 0 is removable iff f(z) is bounded in a punctured neighborhood of z 0. (2) z 0 is a pole of order m iff lim z z0 (z z 0 ) l f(z) = for all l < m, while (z z 0 ) m f(z) has a removable singularity at z 0. Picard s Theorem Let z 0 be an essential singularity of f(z) and U be any punctured neighborhood of z 0, then for any w C, except

perhaps one value, the equation f(z) = w has infinitely many solutions in U. That is to say that an analytic function assumes every complex number, with possibly one exception, as a value in any punctured neighborhood of an essential singularity. as a Singularity If there exist some R > 0 such that f(z) is analytic in the domain { z > R}, then is also called an isolated singularity of f(z). is called a removable singularity, pole or essential singularity of f(z) respectively, if 0 is a removable singularity, pole or essential singularity of f( 1 z ). Meromorphic Function A function f(z) which is analytic in D except for removable singularities and poles, is said to be meromorphic in D. That is to say a meromorphic function has no essential singularity.

A meromorphic function in D can be regarded as a mapping from D to Riemann sphere. Theorem f(z) is a meromorphic function in the extended complex plane C iff f(z) is a rational function.

3.4 The Residue Theorem Definition. If f has an isolated singularity at the point z 0, then the coefficient c 1 of (z z 0 ) 1 in the Laurent expansion for f around z 0 is called the residue of f(z) at z 0 and is denoted by Res(f, z 0 ). If f(z) has a removable singularity at z 0, Res(f, z 0 ) = 0. If f(z) = g(z) h(z) where g(z), h(z) are analytic at z 0, g(z 0 ) 0, h(z 0 ) = 0 and h (z 0 ) 0, then z 0 is a simple pole of f(z) and Res(f, z 0 ) = g(z 0) h (z 0 ). If f(z) has a pole of order k at z 0, then Res(f, z 0 ) = 1 (k 1)! lim z z 0 d k 1 dz k 1[(z z 0) k f(z)].

The residue plays an important role in the contour integration. If Γ is a simple closed positively oriented contour and f(z) is analytic on and inside Γ except for a single isolated singularity z 0 lying interior to Γ, then the integral f(z)dz = 2πiRes(f, z 0 ). In general, we have Γ Cauchy s Residue Theorem. If Γ is a simple closed positively oriented contour and f(z) is analytic on and inside Γ except at the points z 1,..., z n inside Γ, then the integral Γ f(z)dz = 2πi n j=1 Res(f, z j ). Proof. Let D be the interior of Γ and take the disjoint neighborhood B ε (z 1 ), B ε (z 2 ),..., B ε (z n ) in D like the figure,

and set Ω = D \ Γ f(z) dz = Ω n k=1 f(z) dz + B ε (z k ), then n k=1 B ε (z k ) f(z) dz = 2πi n k=1 Res (f, z k ). Residue at. If is an isolated singularity of f(z), i.e. there exists R > 0 such that f(z) is analytic in the annulus {R < z <

}, then the residue of f(z) at is defined by Res(f(z), ) = 1 2πi z =R f(z)dz. Remark. We take the minus sign in the definition, since lies outside the circle { z = R}. If we set w = 1 z then Res(f(z), ) = 1 2πi z =R f(z)dz = 1 2πi w = 1 R f( 1 w ) 1 w 2dw = Res(f( 1 w ) 1 w2, 0). Theorem. If f(z) has finite isolated singularities in the extended complex plane C, the sum of the residues of all the isolated singularities must be zero.

3.5 The Applications of Residue Theorem 1. Trigonometric Integrals over [0, 2π] The real integrals of the form 2π R(cos θ, sin θ)dθ 0 can be evaluated by the residue theorem, where R(cos θ, sin θ) is a rational function (with the real coefficients) of cos θ, sin θ and is finite over [0, 2π]. If we parametrize the unit circle by z = e iθ (0 θ 2π), we have the identities cos θ = 1 ( z + 1 ), sin θ = 1 ( z 1 ), dz = ie iθ dθ. 2 z 2i z

Therefore 2π R(cos θ, sin θ)dθ = 0 z 0 =1 [ ( 1 1 iz R 2 ( z + 1 ), 1 z 2i ( z 1 ))] dz. z

2. Improper Integrals of Certain Functions over (, + ) If P (x), Q(x) are polynomials satisfying deg P (x) deg Q(x) 2, and Q(x) has no real zero on (, + ), then + P (x) dx = 2πi Q(x) Im z 0 >0 Res P (z) Q(z), z 0.

3. Improper Integrals Involving Trigonometric Functions If P (x), Q(x) are polynomials satisfying deg P (x) deg Q(x) 1, and Q(x) has no zero on (, + ), then + + P (x) Q(x) P (x) Q(x) cos αx dx = Re 2πi Im z 0 >0 Res sin αx dx = sgn(α)im 2πi Im z 0 >0 ( ) P (z) Q(z) ei α z, z 0, Res ( ) P (z) Q(z) ei α z, z 0.

Chapter 4 The Applications of Analytic Functions

4.1 Harmonic Functions Definition. A real valued function u(x 1,..., x n ) of n real variables are called harmonic in D if u satisfies Laplace equation at each point of D. u def = 2 u x 2 1 + 2 u x 2 1 + + 2 u x 2 n = 0 Remark. (x 1,..., x n ) must be the euclidean coordinate. Proposition. If f(z) is analytic in D, then both the real part u(x, y) and the imaginary part v(x, y) are harmonic in D. Given a function u(x, y) harmonic in an open disk D, then we can find another harmonic function v(x, y) so that u + iv is analytic in D. Such a function v is called a harmonic conjugate of u.

4.2 Bounds of Analytic Functions The Mean Value Property. Let f be analytic in a domain D and the disc { z z 0 r} be contained in D. Therefore f(z 0 ) = 1 2π 2π 0 f(z 0 + re iθ )dθ. The Maximum Modulus Principle. If f(z) is analytic and nonconstant in a domain D, then f(z) cannot achieve its maximum value in D. Cauchy inequalities. Let f(z) be analytic in a domain D and f(z) M for all z in the disc { z z 0 r} contained in D. Therefore for any nonnegative integer n, f (n) (z 0 ) n!m r n.

The proof can be obtained by using the inequality of integral. Liouville Theorem. constant function. The only bounded entire functions are the Proof. Assume that f(z) M for any z C, then by Cauchy inequalities f (z) M r for any z and positive r. Setting r +, we obtain f (z) = 0 which leads to the conclusion. The Fundamental Theorem of Algebra. Every nonconstant polynomial with complex coefficients has at least one zero. Picard Theorem. Let f(z) be an entire nonconstant function, then for any w C, except perhaps one value, the equation f(z) = w has at least one solution in C.

Applications to Harmonic Functions Theorem. Let u(x, y) be a harmonic function in a simply connected domain D. Then there exist another harmonic function v(x, y) so that f(z) = u + iv is analytic in D. Moreover v is unique up to addition a constant. The Mean Value Property. Let u be harmonic in a domain D and the disc { z z 0 r} be contained in D. Therefore u(x 0, y 0 ) = 1 2π 2π 0 u(x 0 + r cos θ, y 0 + r sin θ)dθ. The Maximum Principle. If u(x, y) is harmonic and nonconstant in a domain D, then u(x, y) can achieve neither maximum nor minimum in D.

Liouville Theorem. If u(x, y) is a bounded harmonic function in R 2, then u(x, y) must be a constant.

Chapter 5 Conformal Mappings

5.1 Geometric Considerations Definition. Assuming that f(z) is analytic at z 0, such a function f(z) is said to be conformal at z 0 if f (z 0 ) 0. A transformation w = f(z), defined in a domain D, is said to be a conformal transformation or conformal mapping, if it is conformal at each point in D. Proposition. If f(z) is conformal at z 0, then f(z) is a one-to-one mapping locally. Remark. f(z) may not be one-to-one globally even if f(z) is conformal mapping. For example, e z is conformal mapping in C, but it is not a one-to-one mapping. Preservation of angles Suppose that C 1, C 2 are two smooth curves passing through z 0 at which f(z) is conformal, and Γ 1 = f(c 1 ), Γ 2 =

f(c 2 ) are the image curves, then the angle from Γ 1 to Γ 2 is the same as the angle from C 1 to C 2, in sense as well as in size. arg f(z 0 ) is the rotation angle. Scale factors Any small line segment with one end point at z 0, where f(z) is conformal, is contracted or expanded in the ratio f (z 0 ) by the mapping. The Open Mapping Property. If f is nonconstant and analytic in a domain D, then its range f(d) is an open set.

Riemann Mapping Theorem. Let D be any simply connected domain in the plane other than the entire plane itself. Then there exists a one-to-one analytic function that maps D onto the open unit disk. Moreover, one can prescribe an arbitrary point of D and a direction through that point which are to be mapped to the origin and the direction of the positive real axis, respectively. Under such restrictions the mapping is unique. Remark. The Riemann mapping theorem is an existence theorem. The precise expression of the Riemann mapping can only be found for some special domains.

5.2 Möbius Transformations Definition. The transformation f(z) = az + b (a, b, c, d C, ad bc 0) cz + d is called a fractional linear transformation or Möbius transformation. Proposition. A composition of two Möbius transformations is a Möbius transform. An inverse transform of a Möbius transform is a Möbius transform. Proposition. A Möbius transformation can be regarded as a oneto-one conformal mapping to the extended complex plane onto itself. Proposition. A Möbius transformation can be written by the composition of three special transformations: (1) Translation

w = z + β; (2) Contraction and rotation w = Az; (3) Inversion w = 1 z. Circles in the extended complex plane We refer circles and lines as circles in the extended complex plane. The circle in the extended complex plane can be represented as the equation Az z + Bz + B z + C = 0 (A, C R, B 2 AC > 0). Circle-preserving Property. Any fractional linear transformation always maps a circle in the extended complex plane to another circle in the extended complex plane. Symmetric pair with respect to a circle. A couple of points z 1, z 2 are called a symmetric pair with respect to a circle C in the extended complex plane if any circle in the extended complex plane passing through z 1, z 2 is perpendicular to C.

Proposition Suppose that z 1, z 2 is a symmetric pair with respect to a circle C in the extended complex plane and f(z) is a Möbius transformation, then f(z 1 ), f(z 2 ) is a symmetric pair with respect to f(c).

5.3 Möbius Transformations, Continued Mapping the upper half plane onto the unit disc w = e iθ z λ z λ. Mapping the unit disc onto the unit disc w = e iθ z α 1 ᾱz. Mapping (z 1, z 2, z 3 ) to (w 1, w 2, w 3 ) z z 1 z z 2 : z 3 z 1 z 3 z 2 = w w 1 w w 2 : w 3 w 1 w 3 w 2.

Each side of the equation above is called the cross ratio.

5.4 The Schwarz-Christoffel Transformation Exponential Function e z is a conformal mapping in C. Power Function z n is a conformal mapping in C \ {0}.

Chapter 6 The Transforms of Applied Mathematics

6.1 The Fourier Transform Definition If f(x) is defined for all real x, then the integral + f(x)e iωx dx (1) is called Fourier transform of f(x), denoted as ˆf(ω) or F [f(x)]. The principle value of the integral 1 2π lim N N + N f(ω)e iωx dω (2) is called inverse Fourier transform of ˇf(ω), denoted by F 1 [f(ω)]. Remark The integral (1) converges for all f L 1 (R), where L 1 (R) = {f : R f dx < + }. Otherwise (1) may diverge.

Fourier Inversion Formula. If f L 1 (R) is continuous except for a finite number of jumps, then lim N + 1 2π N N ˆf(ω)e iωx dω = 1 (f(x + 0) + f(x 0)), 2 where f(x + 0), f(x 0) denote the left and right limits of f at x. Corollary For any f L 1 (R) C(R), f = F 1 (F (f)), where C(R is the set of all the continuous functions defined in R The Properties of Fourier Transform 1. Linearity F (l 1 f 1 + l 2 f 2 ) = l 1 F (f 1 ) + l 2 F (f 2 ). 2. Translation For any x 0 R, F (f(x x 0 )) = e iωx 0 ˆf(ω). 3. Scaling For any nonzero k R, F (f(kx)) = 1 k ˆf( ω k ). 4. Duality If f L 1 (R) C(R) and h(ω) = (f(x))ˆ, then ĥ(ω) = 2πf( ω).

5. Convolution Define the convolution of f 1, f 2 as f 1 f 2 (x) = + f 1 (x y)f 2 (y) dy, then F (f 1 f 2 ) = F (f 1 )F (f 2 ). 6. Differentiation If f, f L 1 (R), then F (f (x)) = iωf (f(x)). 7. Integration F ( x 0 f(τ) dτ) = 1 iω F (f).

6.2 The Laplace Transform Definition If f(t) is defined for t > 0, and the integral 0 f(t)e st dt (3) exists for some s, then the integral (3) is called Laplace transform of f(t), denoted as F (s) or L [f(t)]. Theorem (i) If the integral (3) converges for some real β, then (3) converges whenever Re s > β, and F (s) is analytic in {Re s > β}. (ii) If the integral (3) diverges for some real β, then (3) diverges whenever Re s < β. The Properties of Laplace Transform 1. Linearity L (l 1 f 1 + l 2 f 2 ) = l 1 L (f 1 ) + l 2 L (f 2 ).

2. Time scaling For any positive k, L (f(kt)) = 1 k L (f)(s k ). 3. Time shifting If f(t) = 0 holds for any negative t, then for any a > 0, L (f(t a)) = e as L (f)(s). 4. Frequency shifting L (e at f(t)) = L (f)(s a). 5. Convolution Define the convolution of f 1, f 2 as f 1 f 2 (t) = t 0 f 1 (t τ)f 2 (τ) dτ, then L (f 1 f 2 ) = L (f 1 )L (f 2 ). 6. Differentiation L (f (n) (t)) = s n L (f) s n 1 f(0) f (n 1) (0). 7. Integration L ( t 0 f(τ) dτ) = 1 s L (f). Theorem. For any rational function F (s) = P (s) satisfying deg P (s) < Q(s) deg Q(s), there exists a unique continuous function f(t) such that L (f(t)) = F (s), which is called inverse Laplace transform of F (s), denoted by L 1 (F (s)).

Proof. Since any rational function F (s) = P (s), satisfying deg P (s) < Q(s) deg Q(s), has a partial fraction decomposition we can set F (s) = f(t) = m n j=1 k=1 m n j=1 k=1 C jk (s s j ) k, t k 1 e s jt. We can use Laplace transform to solve linear ODEs (ordinary differential equations).