[Godawari Agarwal, Soma Perera, & Maria Pinelas, 2010] An Introduction to Complex Analysis by Ravi P. A

We also show that the real and imaginary parts of an analytic function are solutions of the Laplace equation. We also show that Rouch'e's Theorem provides locations of the zeros and poles of meromorphic functions.

Complex Numbers I

Let A be a point on the real axis such that OA=a. Sincei·i a=i2a=−a, we can conclude that the double multiplication of the real number abyia amounts to the rotation of OA by two right angles to the positionOA. It naturally follows that the multiplication byiis corresponds to the rotation of OA through one right angle to the position OA. So if yOy is a line perpendicular to the real axis xOx, then all imaginary numbers are represented by points onyOy.

Complex Numbers II

Since the numbering of the terms is arbitrary, the ratio of two non-zero terms must be positive. The number θ is called an argument of z, and we write θ= argz. Geometrically, argz denotes the angle measured in radians that the vector corresponds to tozmakes with the positive real axis.

Complex Numbers III

Now, letz =reiθ =r(cosθ+isinθ). Using the multiplicative property of the exponential function, we get To obtain all the mth roots of z, we must apply formula (3.5) to each polar representation of z. For example, let's find all the roots of unity.

Problems

Similarly, (i) is violated by assuming −i ∈ P. Therefore, the words positive and negative never apply to complex numbers.

Answers or Hints

Set Theory

This is clearly a one-to-one and on (bijective) correspondence between points onS and the extended complex plane. Show that ifz1 and z2 are finite points in the complex plane C, then the distance between their stereographic projection is given by.

Complex Functions

A function f is said to be continuous on the set S if it is continuous at every point. Similarly, a function is continuous if and only if the inverse image of every closed set is closed.

Analytic Functions I

However, uis not continuous in (0,0) and is therefore not differentiable in (0,0). So even if the function f satisfies the Cauchy-Riemann equations at a pointz0, it need not be differentiable at atz0. Letf(z) = u(x, y) +iv(x, y) be defined in an open set S containing the point z0. If the first-order partial derivatives of uenvex exist in S, they are continuous atz0, and satisfy the Cauchy-Riemann equations atz0, thenf is differentiable at z0. Moreover, .

Analytic Functions II

Suppose f = u+iv is analytic in a rectangle with sides parallel to the coordinate axes and satisfies the relationux+vy = 0 for all xandy. Show that there exists a real constant and a complex constant, such that f(z) =−icz+ D. Show that a necessary and sufficient condition for a function f(z) =u(x, y) +iv(x, y) to be analytic in a domain S is that the real part u(x, y) and the imaginary part v(x, y) are conjugate harmonic functions in S.

Elementary Functions I

The functions cotz and cosecz are analytic for all z except at the points z=kπ, while the functions tanz and seczz are analytic for all z except at the points z= (π/2) +kπ, where is an integer. It follows from relations (8.2) that sinhz and cosh are periodic with period 2πi. In addition, the zeros of sinxharez=kπi and the zeros of coshz are z= (k+ 1/2)πi, where is an integer. The functions cothzan and cosech are analytic for all z except at the points z=kπi, while the functions tanhzan and sechzari are analytic for all z except the points z= (k+ 1/2)πi, where is an integer.

Elementary Functions II

The function Logzis analytically in the domain D∗ consisting of all points of the complex plane except those that lie on the non-positive real axis; i.e.,D∗=C−(−∞,0).Furthermore,. Hence, if α is not a real rational number, we get infinitely many different values of zα, one for each choice of the integer (9.4). The inverse sine function w = sin−1z is defined by the equation z = sinw. We will show that sin−1z is a multivalued function given by.

Mappings by Functions I

The image w= (1+i)z+2 transforms the rectangular area in Figure 10.1 to the rectangular area shown in the w-plane. The first of these transformations is an inversion with respect to the unit circle|z|= 1; that is, the image of a nonzero pointz is the point Z with the properties. A line (A= 0) that does not pass through the origin (D= 0) in the z-plane is transformed into a circle through the origin in the w-plane.

Mappings by Functions II

However, we can enlarge the domain of definition of (11.1) to define a linear fractional transformation T on the extended z-plane such that the point w=a/c is the image of z =∞ when c= 0. We first write. When its domain of definition is enlarged in this way, the linear transformation (11.5) is a one-to-one mapping of the extended z-plane onto the extended w-plane. We also note that (11.7) defines the only linear fractional transformation that maps the points z1, z2, and z3 to onow1, w2, and w3, respectively.

Curves, Contours, and Simply Connected Domains

Jordan Curve Theorem). The points on any simple closed curve or simple closed contour γ are boundary points of two

An immediate consequence of Theorem 12.1 is that the interior of a simple closed curve is simply connected. Since I(γ) is bounded, we can take two pointsz1andz2onL\I(γ) such that z0 is on the line segment connecting z1andz2. Then z1 andz2 are in S and z0 is on the line segment connecting them, so z0 ∈S since S is convex, a contradiction.

Complex Integration

Now let S be an open set, and let γ, given by the sequence ofz: [a, b]→C, be a smooth curve in S. Determine the length of the arc of the cycloid given by z(t) =a(t− sint) +a i(1−cost), 0≤t≤2π while is a positive real number. Complex integration 89 13.12.Let γR be the circle|z| be =R described in the counterclockwise direction, where R >0. Suppose Logz is the major branch of the logarithm function.

Independence of Path

At each point of the contourγ, the function 1/zi is the derivative of the main branch of logz. So we have: z−z0)ndz, where nis is an integer not equal to −1 and γr is the circle|z−z0| =r run counterclockwise once. Moreover, (z−z0)n is the derivative of the function (z−z0)n+1/(n+ 1). Since γ is a closed contour lying in S,we. The contour integrals of f are independent of paths in S; i.e. if α, β∈S andγ1 and γ2 are contours in S connecting αandβ, then.

Cauchy-Goursat Theorem

Thus, we have shown that if f is analytic in a singly connected domain and its derivative f(z) is continuous (recall that analyticity guarantees the existence of off(z); but it does not guarantee the continuity of off(z)), then its integral around of any simple closed contour in the domain equal to zero. If γ is a closed contour but not simple, then the integration over γ can always be decomposed into integrations over simple closed curves. Since the interior of a simple closed contour is a simply connected domain, Theorem 15.2 can be stated in a more practical form: If γ is a simple closed contour and f is analytic at every point on and inside γ, then.

Deformation Theorem

Integrandf(z) = (5z−2)/(z2−z) is analytic everywhere except for the zeros in the denominator, z= 0 and z= 1. Let γ1andγ2 be two small circles that enclose these points. For each of the following functionsf, describe the domain of analysis and use the Cauchy-Goursat theorem to show it. Assuming that γ is oriented such that the points of the domain lie to the left of γ, it shows.

Cauchy’s Integral Formula

The first two terms in the equation above are independent of r, and so the value of the last term does not change if we let→0; i.e., Now we will prove the following general result, which we will need in the next lecture. Let M = maxξ∈γ|g(ξ)|and let equal to the shortest distance fromztoγ, so that |ξ−z| ≥d >0 for all ξonγ. As we let Δz approach zero, we can assume that|Δz|< d/2. Then, by the triangle inequality, we have

Cauchy’s Integral Formula for Derivatives

Morera’s Theorem). If f is continuous in a domain S and

Let γRof be an analytic function inside and on a circle radiusCentered at z0.If|f(z)| The function ≤M inside and on a circle radiusγRofCentered at z0.If|f(z)| ≤M for allzonγR, then the following inequality holds. 18.6) ForzonγR, the integrand is bounded by M/Rn+1 and the length of γR is 2πR. Thus, from theorem 13.1, it follows that.

Liouville’s Theorem). The only bounded entire functions are the constant functions

Let f be analytic inside and on a positively oriented closed contourγ, and the pointz0 is not onγ. show it Letu=u(x, y) be a harmonic function in a domainS. Show that all partial derivatives ux, uy, uxx, uxy, uyy,· · · exist and are harmonic. Let f(z, λ) be an analytic function of z on the domain S for every value ofλin|λ−λ0|< ρ, andf(z, λ)→F(z) uniformly asλ→λ0 in every closed regionGof S .Show that F(z) is analytic on S.

The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra)

Let A = max

From (19.6) we can conclude that Pn(z) does not vanish for anyz=zj, i≤j ≤n. Furthermore, the factorization (19.5) is unique except for the order of factors. By multiplying the factors in (19.6) and equating the coecients of identical powers of the left and right sides, we obtain the following relations between the roots and the coecients of Pn(z).

Let B = max

We will prove that “If all zeros of a polynomialPn(z) lie in a half-plane, then all zeros of the derivativePn(z) lie in the same half-plane.”. The general result will follow by considering the half-faces below (above) each pair of adjacent vertices of the smallest convex polygon containing the zeros of Pn(z). This follows from (19.6). Show that if the coefficients of the polynomial equation Pn(z) = 0 are real and ifz0 is a root, thenz0 is also a root.

Maximum Modulus Principle

Minimum Modulus Principle). A function f analytic in a bounded domain S, continuous up to and including its

Assume that f is analytic in a bounded domain S, continuous up to and including its boundary, and f(z)= 0 for allz ∈ S. Then |f| a maximum M and a minimum mon the limit, and either f is a constant or m <|f(z)|< M for allz∈S. Consider the function g(z) defined in (18.7) on the rectangle with vertices at 0, π, i, π+i. Find the maximum and minimum values of|g(z)| and determine where these values occur.

Sequences and Series of Numbers

A point z is called a limit point (or accumulation point) of the sequence {zn} if for every neighborhood N(z) there exists a subsequence {znk} for which all terms belong to N(z). If the sequence of partial sums {sn}∞n=0 has a limit s, the series is said to converge, orsum, to s, and we write=$∞. j=0ms. A series that does not converge is said to diverge. j=0zj is said to be absolutely convergent provided that the series of quantities.

Sequences and Series of Functions

The series $ ∞

If the series $ ∞

Moreover, for every k≥1, the diﬀerentiated sequence {fn(k)(z)} converges to f(k)(z) uniformly on every compact subset S. The sequence {(sinnx)/n} converges uniformly to zero on the real axis; however, the sequence of its derivative {cosnx} converges only at x= 0. Thus, the sequence {(sinnz)/n} cannot converge uniformly on any domain containing points of the real axis. It is clear that the functional+m(z)−sn(z), which is a finite sum of analytic functions, is analytic on S and continuous on S.

Power Series

From Theorem 23.1 and Corollary 23.1, it is clear that the radius of convergenceRof (23.1) is the smallest upper limit for the distances|z−z0|from pointz0 to pointzat, which the series (23.1) converges. If the sum of two power series near the expansion point z0 is the same, then show that the identical powers of (z−z0) have identical coefficients; i.e., there is a unique power series that has a given sum in a quarter of z0.

Taylor’s Series

A function f(z) is analytic atz0 if and only if it can be expanded into a Taylor series atz0. By calculating the derivatives of all types at z0= 0 of the entire functions z, cosz, sinz, coshz, sinhz, we obtain the following expansions of the Maclaurin series that apply to|z|<∞:. This is true for |z−1|<1, the largest open disk with center 1 over which Logzis is analytic.

Laurent’s Series

Laurent's series of an analytic function in an annular region can be differentiated term-by-term. As a result, since Logz is not analytic in any annulus around 0, it cannot be represented by a Laurent series around 0.

Zeros of Analytic Functions

We claim that f(z) is identical to zero in some neighborhood α. Indeed, if (z)= 0 in some punctured neighborhood N(α), then by definition. It follows from Corollary 26.6 that an analytic function can have an infinite number of zeros only in an open or unbounded domain. It also follows from Corollary 26.6 that the entire function in any bounded part of the complex plane can have only a finite number of zeros.

Analytic Continuation

Then in some domain S⊆C such that (a, b)⊂S, there may exist a unique analytic function f(z), z∈S which coincides with f(x) on (a, b). This function f(z) is also called an analytic continuation. functions f(x) of a real variable into the complex domain S. Various such analytic continuations were given in Lecture 26. Within its circle of convergence, which is the unit circle centered at the point z=−i, the series is convergent.