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Complex Variables For ECON 397 Macroeconometrics Steve Cunningham

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Open Disks or Neighborhoods Definition. The set of all points z which satisfy the inequality |z – z 0 |<, where is a positive real number is called an open disk or neighborhood of z 0. Definition. The set of all points z which satisfy the inequality |z – z 0 |<, where is a positive real number is called an open disk or neighborhood of z 0. Remark. The unit disk, i.e., the neighborhood |z|< 1, is of particular significance. Remark. The unit disk, i.e., the neighborhood |z|< 1, is of particular significance. 1

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Interior Point Definition. A point is called an interior point of S if and only if there exists at least one neighborhood of z 0 which is completely contained in S. Definition. A point is called an interior point of S if and only if there exists at least one neighborhood of z 0 which is completely contained in S. z0z0 S

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Open Set. Closed Set. Definition. If every point of a set S is an interior point of S, we say that S is an open set. Definition. If every point of a set S is an interior point of S, we say that S is an open set. Definition. If B(S) S, i.e., if S contains all of its boundary points, then it is called a closed set. Definition. If B(S) S, i.e., if S contains all of its boundary points, then it is called a closed set. Sets may be neither open nor closed. Sets may be neither open nor closed. Open Closed Neither

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Connected An open set S is said to be connected if every pair of points z 1 and z 2 in S can be joined by a polygonal line that lies entirely in S. Roughly speaking, this means that S consists of a single piece, although it may contain holes. An open set S is said to be connected if every pair of points z 1 and z 2 in S can be joined by a polygonal line that lies entirely in S. Roughly speaking, this means that S consists of a single piece, although it may contain holes. S z1z1 z2z2

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Domain, Region, Closure, Bounded, Compact An open, connected set is called a domain. A region is a domain together with some, none, or all of its boundary points. The closure of a set S denoted, is the set of S together with all of its boundary. Thus. An open, connected set is called a domain. A region is a domain together with some, none, or all of its boundary points. The closure of a set S denoted, is the set of S together with all of its boundary. Thus. A set of points S is bounded if there exists a positive real number R such that |z|

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Review: Real Functions of Real Variables Definition. Let. A function f is a rule which assigns to each element a one and only one element b,. We write f:, or in the specific case b = f(a), and call b the image of a under f. We call the domain of definition of f or simply the domain of f. We call the range of f. We call the set of all the images of, denoted f ( ), the image of the function f. We alternately call f a mapping from to. Definition. Let. A function f is a rule which assigns to each element a one and only one element b,. We write f:, or in the specific case b = f(a), and call b the image of a under f. We call the domain of definition of f or simply the domain of f. We call the range of f. We call the set of all the images of, denoted f ( ), the image of the function f. We alternately call f a mapping from to.

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Real Function In effect, a function of a real variable maps from one real line to another. In effect, a function of a real variable maps from one real line to another. f

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Complex Function Definition. Complex function of a complex variable. Let C. A function f defined on is a rule which assigns to each z a complex number w. The number w is called a value of f at z and is denoted by f(z), i.e., w = f(z). The set is called the domain of definition of f. Although the domain of definition is often a domain, it need not be. Definition. Complex function of a complex variable. Let C. A function f defined on is a rule which assigns to each z a complex number w. The number w is called a value of f at z and is denoted by f(z), i.e., w = f(z). The set is called the domain of definition of f. Although the domain of definition is often a domain, it need not be.

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Remark Properties of a real-valued function of a real variable are often exhibited by the graph of the function. But when w = f(z), where z and w are complex, no such convenient graphical representation is available because each of the numbers z and w is located in a plane rather than a line. Properties of a real-valued function of a real variable are often exhibited by the graph of the function. But when w = f(z), where z and w are complex, no such convenient graphical representation is available because each of the numbers z and w is located in a plane rather than a line. We can display some information about the function by indicating pairs of corresponding points z = (x,y) and w = (u,v). To do this, it is usually easiest to draw the z and w planes separately. We can display some information about the function by indicating pairs of corresponding points z = (x,y) and w = (u,v). To do this, it is usually easiest to draw the z and w planes separately.

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Graph of Complex Function xu yv z- plane w- plane domain of definition range w = f(z)

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Example 1 Describe the range of the function f(z) = x 2 + 2i, defined on (the domain is) the unit disk |z| 1. Solution: We have u(x,y) = x 2 and v(x,y) = 2. Thus as z varies over the closed unit disk, u varies between 0 and 1, and v is constant (=2). Therefore w = f(z) = u(x,y) + iv(x,y) = x 2 +2i is a line segment from w = 2i to w = 1 + 2i. x y u v f(z) domain range

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Example 2 Describe the function f(z) = z 3 for z in the semidisk given by |z| 2, Im z 0. Solution: We know that the points in the sector of the semidisk from Arg z = 0 to Arg z = 2 /3, when cubed cover the entire disk |w| 8 because The cubes of the remaining points of z also fall into this disk, overlapping it in the upper half- plane as depicted on the next screen.

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2-2 x y u v w = z 3

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Sequence Definition. A sequence of complex numbers, denoted, is a function f, such that f: N C, i.e, it is a function whose domain is the set of natural numbers between 1 and k, and whose range is a subset of the complex numbers. If k =, then the sequence is called infinite and is denoted by, or more often, z n. (The notation f(n) is equivalent.) Definition. A sequence of complex numbers, denoted, is a function f, such that f: N C, i.e, it is a function whose domain is the set of natural numbers between 1 and k, and whose range is a subset of the complex numbers. If k =, then the sequence is called infinite and is denoted by, or more often, z n. (The notation f(n) is equivalent.) Having defined sequences and a means for measuring the distance between points, we proceed to define the limit of a sequence. Having defined sequences and a means for measuring the distance between points, we proceed to define the limit of a sequence.

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Limit of a Sequence Definition. A sequence of complex numbers is said to have the limit z 0, or to converge to z 0, if for any > 0, there exists an integer N such that |z n – z 0 | N. We denote this by Definition. A sequence of complex numbers is said to have the limit z 0, or to converge to z 0, if for any > 0, there exists an integer N such that |z n – z 0 | N. We denote this by Geometrically, this amounts to the fact that z 0 is the only point of z n such that any neighborhood about it, no matter how small, contains an infinite number of points z n. Geometrically, this amounts to the fact that z 0 is the only point of z n such that any neighborhood about it, no matter how small, contains an infinite number of points z n.

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Limit of a Function We say that the complex number w 0 is the limit of the function f(z) as z approaches z 0 if f(z) stays close to w 0 whenever z is sufficiently near z 0. Formally, we state: We say that the complex number w 0 is the limit of the function f(z) as z approaches z 0 if f(z) stays close to w 0 whenever z is sufficiently near z 0. Formally, we state: Definition. Limit of a Complex Sequence. Let f(z) be a function defined in some neighborhood of z 0 except with the possible exception of the point z 0 is the number w 0 if for any real number > 0 there exists a positive real number > 0 such that |f(z) – w 0 | 0 there exists a positive real number > 0 such that |f(z) – w 0 |< whenever 0<|z - z 0 |<.

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Limits: Interpretation of z 0, we can find a corresponding disk about w 0 such that all the points in the disk about z 0 are mapped into it. That is, any neighborhood of w 0 contains all the values assumed by f in some full neighborhood of z 0, except possibly f(z 0 ). We can interpret this to mean that if we observe points z within a radius of z 0, we can find a corresponding disk about w 0 such that all the points in the disk about z 0 are mapped into it. That is, any neighborhood of w 0 contains all the values assumed by f in some full neighborhood of z 0, except possibly f(z 0 ). z0z0z0z0 w0w0 w = f(z) z-plane w-plane u v x y

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Properties of Limits If as z z 0, lim f(z) A and lim g(z) B, then lim [ f(z) g(z) ] = A B lim [ f(z) g(z) ] = A B lim f(z)g(z) = AB, and lim f(z)g(z) = AB, and lim f(z)/g(z) = A/B. if B 0. lim f(z)/g(z) = A/B. if B 0.

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Continuity Definition. Let f(z) be a function such that f: C C. We call f(z) continuous at z 0 iff: Definition. Let f(z) be a function such that f: C C. We call f(z) continuous at z 0 iff: F is defined in a neighborhood of z 0, F is defined in a neighborhood of z 0, The limit exists, and The limit exists, and A function f is said to be continuous on a set S if it is continuous at each point of S. If a function is not continuous at a point, then it is said to be singular at the point. A function f is said to be continuous on a set S if it is continuous at each point of S. If a function is not continuous at a point, then it is said to be singular at the point.

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Note on Continuity One can show that f(z) approaches a limit precisely when its real and imaginary parts approach limits, and the continuity of f(z) is equivalent to the continuity of its real and imaginary parts. One can show that f(z) approaches a limit precisely when its real and imaginary parts approach limits, and the continuity of f(z) is equivalent to the continuity of its real and imaginary parts.

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Properties of Continuous Functions If f(z) and g(z) are continuous at z 0, then so are f(z) g(z) and f(z)g(z). The quotient f(z)/g(z) is also continuous at z 0 provided that g(z 0 ) 0. If f(z) and g(z) are continuous at z 0, then so are f(z) g(z) and f(z)g(z). The quotient f(z)/g(z) is also continuous at z 0 provided that g(z 0 ) 0. Also, continuous functions map compact sets into compact sets. Also, continuous functions map compact sets into compact sets.

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Derivatives Differentiation of complex-valued functions is completely analogous to the real case: Differentiation of complex-valued functions is completely analogous to the real case: Definition. Derivative. Let f(z) be a complex- valued function defined in a neighborhood of z 0. Then the derivative of f(z) at z 0 is given by Provided this limit exists. F(z) is said to be differentiable at z 0. Definition. Derivative. Let f(z) be a complex- valued function defined in a neighborhood of z 0. Then the derivative of f(z) at z 0 is given by Provided this limit exists. F(z) is said to be differentiable at z 0.

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Properties of Derivatives

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Analytic. Holomorphic. Definition. A complex-valued function f (z) is said to be analytic, or equivalently, holomorphic, on an open set if it has a derivative at every point of. (The term regular is also used.) Definition. A complex-valued function f (z) is said to be analytic, or equivalently, holomorphic, on an open set if it has a derivative at every point of. (The term regular is also used.) It is important that a function may be differentiable at a single point only. Analyticity implies differentiability within a neighborhood of the point. This permits expansion of the function by a Taylor series about the point. It is important that a function may be differentiable at a single point only. Analyticity implies differentiability within a neighborhood of the point. This permits expansion of the function by a Taylor series about the point. If f (z) is analytic on the whole complex plane, then it is said to be an entire function. If f (z) is analytic on the whole complex plane, then it is said to be an entire function.

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Rational Function. Definition. If f and g are polynomials in z, then h (z) = f (z)/g(z), g(z) 0 is called a rational function. Definition. If f and g are polynomials in z, then h (z) = f (z)/g(z), g(z) 0 is called a rational function. Remarks. Remarks. All polynomial functions of z are entire. All polynomial functions of z are entire. A rational function of z is analytic at every point for which its denominator is nonzero. A rational function of z is analytic at every point for which its denominator is nonzero. If a function can be reduced to a polynomial function which does not involve, then it is analytic. If a function can be reduced to a polynomial function which does not involve, then it is analytic.

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Example 1 Thus f 1 (z) is analytic at all points except z=1.

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Example 2 Thus f 2 (z) is nowhere analytic.

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Testing for Analyticity Determining the analyticity of a function by searching for in its expression that cannot be removed is at best awkward. Observe: It would be difficult and time consuming to try to reduce this expression to a form in which you could be sure that the could not be removed. The method cannot be used when anything but algebraic functions are used.

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Cauchy-Riemann Equations (1) If the function f (z) = u(x,y) + iv(x,y) is differentiable at z 0 = x 0 + iy 0, then the limit can be evaluated by allowing z to approach zero from any direction in the complex plane.

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Cauchy-Riemann Equations (2) If it approaches along the x-axis, then z = x, and we obtain But the limits of the bracketed expression are just the first partial derivatives of u and v with respect to x, so that:

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Cauchy-Riemann Equations (3) If it approaches along the y-axis, then z = y, and we obtain And, therefore

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Cauchy-Riemann Equations (4) By definition, a limit exists only if it is unique. Therefore, these two expressions must be equivalent. Equating real and imaginary parts, we have that must hold at z 0 = x 0 + iy 0. These equations are called the Cauchy-Riemann Equations. Their importance is made clear in the following theorem.

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Cauchy-Riemann Equations (5) Theorem. Let f (z) = u(x,y) + iv(x,y) be defined in some open set containing the point z 0. If the first partial derivatives of u and v exist in, and are continuous at z 0, and satisfy the Cauchy-Riemann equations at z 0, then f (z) is differentiable at z 0. Consequently, if the first partial derivatives are continuous and satisfy the Cauchy-Riemann equations at all points of, then f (z) is analytic in. Theorem. Let f (z) = u(x,y) + iv(x,y) be defined in some open set containing the point z 0. If the first partial derivatives of u and v exist in, and are continuous at z 0, and satisfy the Cauchy-Riemann equations at z 0, then f (z) is differentiable at z 0. Consequently, if the first partial derivatives are continuous and satisfy the Cauchy-Riemann equations at all points of, then f (z) is analytic in.

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Example 1 Hence, the Cauchy-Riemann equations are satisfied only on the line x = y, and therefore in no open disk. Thus, by the theorem, f (z) is nowhere analytic.

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Example 2 Prove that f (z) is entire and find its derivative. The first partials are continuous and satisfy the Cauchy-Riemann equations at every point.

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Harmonic Functions Definition. Harmonic. A real-valued function (x,y) is said to be harmonic in a domain D if all of its second-order partial derivatives are continuous in D and if each point of D satisfies Definition. Harmonic. A real-valued function (x,y) is said to be harmonic in a domain D if all of its second-order partial derivatives are continuous in D and if each point of D satisfies Theorem. If f (z) = u(x,y) + iv(x,y) is analytic in a domain D, then each of the functions u(x,y) and v(x,y) is harmonic in D.

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Harmonic Conjugate Given a function u(x,y) harmonic in, say, an open disk, then we can find another harmonic function v(x,y) so that u + iv is an analytic function of z in the disk. Such a function v is called a harmonic conjugate of u. Given a function u(x,y) harmonic in, say, an open disk, then we can find another harmonic function v(x,y) so that u + iv is an analytic function of z in the disk. Such a function v is called a harmonic conjugate of u.

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Example Construct an analytic function whose real part is: Solution: First verify that this function is harmonic.

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Example, Continued Integrate (1) with respect to y:

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Example, Continued Now take the derivative of v(x,y) with respect to x: According to equation (2), this equals 6xy – 1. Thus,

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Example, Continued The desired analytic function f (z) = u + iv is:

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Complex Exponential We would like the complex exponential to be a natural extension of the real case, with f (z) = e z entire. We begin by examining e z = e x+iy = e x e iy. We would like the complex exponential to be a natural extension of the real case, with f (z) = e z entire. We begin by examining e z = e x+iy = e x e iy. e iy = cos y + i sin y by Eulers and DeMoivres relations. e iy = cos y + i sin y by Eulers and DeMoivres relations. Definition. Complex Exponential Function. If z = x + iy, then e z = e x (cos y + i sin y). Definition. Complex Exponential Function. If z = x + iy, then e z = e x (cos y + i sin y). That is, |e z |= e x and arg e z = y. That is, |e z |= e x and arg e z = y.

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More on Exponentials Recall that a function f is one-to-one on a set S if the equation f (z 1 ) = f (z 2 ), where z 1, z 2 S, implies that z 1 = z 2. The complex exponential function is not one-to-one on the whole plane. Recall that a function f is one-to-one on a set S if the equation f (z 1 ) = f (z 2 ), where z 1, z 2 S, implies that z 1 = z 2. The complex exponential function is not one-to-one on the whole plane. Theorem. A necessary and sufficient condition that e z = 1 is that z = 2k i, where k is an integer. Also, a necessary and sufficient condition that is that z 1 = z 2 + 2k i, where k is an integer. Thus e z is a periodic function. Theorem. A necessary and sufficient condition that e z = 1 is that z = 2k i, where k is an integer. Also, a necessary and sufficient condition that is that z 1 = z 2 + 2k i, where k is an integer. Thus e z is a periodic function.

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