Dr. John Mathews: New Applications
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en-us2016 Maplesoft, A Division of Waterloo Maple Inc.Maplesoft Document SystemMon, 02 May 2016 23:14:15 GMTMon, 02 May 2016 23:14:15 GMTNew applications published by Dr. John Mathewshttp://www.mapleprimes.com/images/mapleapps.gifDr. John Mathews: New Applications
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The Origin of Complex Numbers
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The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.<img src="/applications/images/app_image_blank_lg.jpg" alt="The Origin of Complex Numbers" align="left"/>The origin of complex numbers starts with the contributions of Scipione del Ferro, Nicolo Tartaglia, Girolamo Cardano, and Rafael Bombelli. This Maple worksheed details the methods and formulas they used. It explores these formulas using Maple and shows how they can be extended. Numerous examples, exercises and illustrations make this a useful teaching module for an introduction of complex numbers.126618Fri, 14 Oct 2011 04:00:00 ZDr. John MathewsDr. John MathewsComplex Analysis Project
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Complex analysis Maple 12 worksheets to accompany the textbook: COMPLEX ANALYSIS: for Mathematics and Engineering, Fifth Edition, 2006
by John H. Mathews and Russell W. Howell
ISBN: 0-7637-3748-8
Jones and Bartlett Pub. Inc.<img src="/view.aspx?si=4846/0763737488.01._AA240_SCLZZZZZZZ_.jpg" alt="Complex Analysis Project" align="left"/>Complex analysis Maple 12 worksheets to accompany the textbook: COMPLEX ANALYSIS: for Mathematics and Engineering, Fifth Edition, 2006
by John H. Mathews and Russell W. Howell
ISBN: 0-7637-3748-8
Jones and Bartlett Pub. Inc.4846Mon, 27 Nov 2006 00:00:00 ZDr. John MathewsDr. John MathewsSection 5.3 Complex Exponents
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In Section 1.5 we indicated that the complex numbers are complete in the sense that it is possible to make sense out of expressions such as sqrt(1+i); or (-1)^i; left without appealing to a number system beyond the framework of complex numbers. We will do this by taking note of some rudimentary properties of the complex exponential and logarithm, and then using our imagination.<img src="/view.aspx?si=4619//applications/images/app_image_blank_lg.jpg" alt="Section 5.3 Complex Exponents" align="left"/>In Section 1.5 we indicated that the complex numbers are complete in the sense that it is possible to make sense out of expressions such as sqrt(1+i); or (-1)^i; left without appealing to a number system beyond the framework of complex numbers. We will do this by taking note of some rudimentary properties of the complex exponential and logarithm, and then using our imagination.4619Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 5.2 Branches of the Complex Logarithm Function
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In Section 5.1 we showed that if w; is a nonzero complex number the equation w = exp(z); has infinitely many solutions. Because the function `exp(z)`; is a many-to-one function, its inverse (the logarithm) is necessarily multivalued.<img src="/view.aspx?si=4618//applications/images/app_image_blank_lg.jpg" alt="Section 5.2 Branches of the Complex Logarithm Function" align="left"/>In Section 5.1 we showed that if w; is a nonzero complex number the equation w = exp(z); has infinitely many solutions. Because the function `exp(z)`; is a many-to-one function, its inverse (the logarithm) is necessarily multivalued.4618Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 5.1 The Complex Exponential Function
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How should complex-valued functions such as exp(z);, log(z);, sin(z);, etc., be defined? Clearly, any responsible definition should satisfy the following criteria:
i. The functions so defined must give the same values as the corresponding functions for real variables when the number z; is a real number.
ii. As far as possible, the properties of these new functions must correspond with their real counterparts. For example, we would want
exp(z[1]+z[2]) = exp(z[1])*exp(z[2]);
to be valid regardless of whether z[1]; and z[2]; were real or complex.<img src="/view.aspx?si=4617//applications/images/app_image_blank_lg.jpg" alt="Section 5.1 The Complex Exponential Function" align="left"/>How should complex-valued functions such as exp(z);, log(z);, sin(z);, etc., be defined? Clearly, any responsible definition should satisfy the following criteria:
i. The functions so defined must give the same values as the corresponding functions for real variables when the number z; is a real number.
ii. As far as possible, the properties of these new functions must correspond with their real counterparts. For example, we would want
exp(z[1]+z[2]) = exp(z[1])*exp(z[2]);
to be valid regardless of whether z[1]; and z[2]; were real or complex.4617Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 4.4 Power Series Functions
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The function f(z) = sum(a[n]*z^n, n=0..infinity) is called a power series.<img src="/view.aspx?si=4616//applications/images/app_image_blank_lg.jpg" alt="Section 4.4 Power Series Functions" align="left"/>The function f(z) = sum(a[n]*z^n, n=0..infinity) is called a power series.4616Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 4.2 Julia and Mandelbrot Sets
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The Julia set associated to a complex number c is found by iterating the map z = z^2 + c.
The set of points that do not escape to infinity is the Julia set.<img src="/view.aspx?si=4614//applications/images/app_image_blank_lg.jpg" alt="Section 4.2 Julia and Mandelbrot Sets" align="left"/>The Julia set associated to a complex number c is found by iterating the map z = z^2 + c.
The set of points that do not escape to infinity is the Julia set.4614Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 4.1 Sequences and Series
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In formal terms, a complex sequence is a function whose domain is the positive integers and whose range is a subset of the complex numbers. For convenience, we at times use the term sequence rather than complex sequence.
If we wish a function s; to represent an arbitrary sequence, we could specify it by writing s(1) = z[1];, s(2) = z[2];, and so on. The values z[1], z[2], z[3];, ..., are called the terms of a sequence, and mathematicians, being generally lazy when it comes to things like this, often refer to z[1], z[2], z[3];, etc., as the sequence itself, even though they are really speaking of the range of the sequence when they do this. Mathematicians are also not so fussy about starting a sequence at z[1], so that z[0];, z[1];, z[2];, ..., etc., would also be acceptable notation, provided all terms were defined.<img src="/view.aspx?si=4613//applications/images/app_image_blank_lg.jpg" alt="Section 4.1 Sequences and Series" align="left"/>In formal terms, a complex sequence is a function whose domain is the positive integers and whose range is a subset of the complex numbers. For convenience, we at times use the term sequence rather than complex sequence.
If we wish a function s; to represent an arbitrary sequence, we could specify it by writing s(1) = z[1];, s(2) = z[2];, and so on. The values z[1], z[2], z[3];, ..., are called the terms of a sequence, and mathematicians, being generally lazy when it comes to things like this, often refer to z[1], z[2], z[3];, etc., as the sequence itself, even though they are really speaking of the range of the sequence when they do this. Mathematicians are also not so fussy about starting a sequence at z[1], so that z[0];, z[1];, z[2];, ..., etc., would also be acceptable notation, provided all terms were defined.4613Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 3.3 Harmonic Functions
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Let phi(x,y); be a real-valued function of the two real variables x; and y;. The partial differential equation
phi[xx](x,y)+phi[yy](x,y) = 0;
is known as Laplace's equation and is sometimes referred to as the potential equation. If phi;, phi[x];, phi[y];, phi[xx];, phi[xy];, and phi[yy]; are all continuous and if phi(x,y) satisfies Laplace's equation, then phi(x,y) is called a harmonic function. Harmonic functions are important in the areas of applied mathematics, engineering, and mathematical physics. They are used to solve problems involving steady state temperatures, two-dimensional electrostatics, and ideal fluid flow. In Chapter 10 we will see how complex analysis techniques can be used to solve some problems involving harmonic functions. We begin with an important theorem relating analytic and harmonic functions.<img src="/view.aspx?si=4612//applications/images/app_image_blank_lg.jpg" alt="Section 3.3 Harmonic Functions" align="left"/>Let phi(x,y); be a real-valued function of the two real variables x; and y;. The partial differential equation
phi[xx](x,y)+phi[yy](x,y) = 0;
is known as Laplace's equation and is sometimes referred to as the potential equation. If phi;, phi[x];, phi[y];, phi[xx];, phi[xy];, and phi[yy]; are all continuous and if phi(x,y) satisfies Laplace's equation, then phi(x,y) is called a harmonic function. Harmonic functions are important in the areas of applied mathematics, engineering, and mathematical physics. They are used to solve problems involving steady state temperatures, two-dimensional electrostatics, and ideal fluid flow. In Chapter 10 we will see how complex analysis techniques can be used to solve some problems involving harmonic functions. We begin with an important theorem relating analytic and harmonic functions.4612Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 3.2 The Cauchy-Riemann Equations
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We saw in the last section that computing the derivative of complex functions written in a nice form such as f(z) = z^2; is a rather simple task. But life is not so easy, for many times we encounter complex functions written as f(x+i*y) = u(x,y)+i*v(x,y);. For example, suppose we had
f(x+i*y) = x^3-3*x*y^2+i*(3*x^2*y-y^3).
Is there some criterion---perhaps involving the partial derivatives for u;, and v; - - that we can use to determine whether f; is differentiable, and if so, to find the value of `f '(z)`;?<img src="/view.aspx?si=4611//applications/images/app_image_blank_lg.jpg" alt="Section 3.2 The Cauchy-Riemann Equations" align="left"/>We saw in the last section that computing the derivative of complex functions written in a nice form such as f(z) = z^2; is a rather simple task. But life is not so easy, for many times we encounter complex functions written as f(x+i*y) = u(x,y)+i*v(x,y);. For example, suppose we had
f(x+i*y) = x^3-3*x*y^2+i*(3*x^2*y-y^3).
Is there some criterion---perhaps involving the partial derivatives for u;, and v; - - that we can use to determine whether f; is differentiable, and if so, to find the value of `f '(z)`;?4611Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 3.1 Differentiable Functions
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Does the notion of a derivative of a complex function make sense? If so, how should it be defined, and what does it represent? These and other questions will be the focus of the next few sections.
Using our imagination, we take our lead from elementary Calculus and define the derivate of f; at z[0];, written f '(z[0];), by
f '(z[0];) = Limit((f(z)-f(z[0]))/(z-z[0]),z = z[0]); ,
provided that the limit exists. When this happens, we say that the function f; is differentiable at z[0];. If we write Delta*z = z-z[0];, then this definition can be expressed in the form
f '(z[0];) = Limit((f(z[0]+Delta*z)-f(z[0]))/(Delta*z),Delta*z = 0) .<img src="/view.aspx?si=4610//applications/images/app_image_blank_lg.jpg" alt="Section 3.1 Differentiable Functions" align="left"/>Does the notion of a derivative of a complex function make sense? If so, how should it be defined, and what does it represent? These and other questions will be the focus of the next few sections.
Using our imagination, we take our lead from elementary Calculus and define the derivate of f; at z[0];, written f '(z[0];), by
f '(z[0];) = Limit((f(z)-f(z[0]))/(z-z[0]),z = z[0]); ,
provided that the limit exists. When this happens, we say that the function f; is differentiable at z[0];. If we write Delta*z = z-z[0];, then this definition can be expressed in the form
f '(z[0];) = Limit((f(z[0]+Delta*z)-f(z[0]))/(Delta*z),Delta*z = 0) .4610Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 2.6 The Reciprocal Transformation w = 1/z
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The mapping w = 1/z is called the reciprocal transformation and maps the z; plane one-to-one and onto the w; plane except for the point z = 0;, which has no image, and the point w = 0;, which has no preimage or inverse image. Using exponential notation w = rho*exp(i*phi);, we see that if z; = r*exp(i*theta) <> 0;, then we have
w; = rho*exp(i*phi) = 1/z = 1/r; exp(i*theta);.
It is convenient to extend the system of complex numbers by joining to it an "ideal" point denoted by infinity; and called the point at infinity. This new set is called the extended complex plane. The reciprocal transformation maps the "extended complex z-plane" one-to-one and onto the "extended complex w-plane"<img src="/view.aspx?si=4609//applications/images/app_image_blank_lg.jpg" alt="Section 2.6 The Reciprocal Transformation w = 1/z" align="left"/>The mapping w = 1/z is called the reciprocal transformation and maps the z; plane one-to-one and onto the w; plane except for the point z = 0;, which has no image, and the point w = 0;, which has no preimage or inverse image. Using exponential notation w = rho*exp(i*phi);, we see that if z; = r*exp(i*theta) <> 0;, then we have
w; = rho*exp(i*phi) = 1/z = 1/r; exp(i*theta);.
It is convenient to extend the system of complex numbers by joining to it an "ideal" point denoted by infinity; and called the point at infinity. This new set is called the extended complex plane. The reciprocal transformation maps the "extended complex z-plane" one-to-one and onto the "extended complex w-plane"4609Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 2.5 Branches of Functions
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In Section 2.3 we defined the principal square root function and investigated some of its properties. We left some unanswered questions concerning the choices of square roots. We now look into this problem because it is similar to situations involving other elementary functions.<img src="/view.aspx?si=4608//applications/images/app_image_blank_lg.jpg" alt="Section 2.5 Branches of Functions" align="left"/>In Section 2.3 we defined the principal square root function and investigated some of its properties. We left some unanswered questions concerning the choices of square roots. We now look into this problem because it is similar to situations involving other elementary functions.4608Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 2.4 Limits and Continuity
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Let u = u(x, y) be a real-valued function of the two real variables x and y . Recall that u has the u[0] as (x, y ) approaches (x[0], y[0] ) provided that the value of u(x, y) can be made to get as close as we please to the value u(x[0], y[0]) by taking (x, y ) to be sufficiently close to (x[0], y[0] ).<img src="/view.aspx?si=4607//applications/images/app_image_blank_lg.jpg" alt="Section 2.4 Limits and Continuity" align="left"/>Let u = u(x, y) be a real-valued function of the two real variables x and y . Recall that u has the u[0] as (x, y ) approaches (x[0], y[0] ) provided that the value of u(x, y) can be made to get as close as we please to the value u(x[0], y[0]) by taking (x, y ) to be sufficiently close to (x[0], y[0] ).4607Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 2.3 The Mappings w = z^n and w = z^`1/n`
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The mapping w = z^2 or w = x^2-y^2+i*2*x*y
can be expressed in polar coordinates by the function f(z) = r^2*exp(i*2*theta) .
The mapping w = sqrt(z) can be expressed in polar coordinates
by the function f(z) = f(r*exp(i*theta)) = sqrt(r)*exp(i*theta/2) .<img src="/view.aspx?si=4606//applications/images/app_image_blank_lg.jpg" alt="Section 2.3 The Mappings w = z^n and w = z^`1/n`" align="left"/>The mapping w = z^2 or w = x^2-y^2+i*2*x*y
can be expressed in polar coordinates by the function f(z) = r^2*exp(i*2*theta) .
The mapping w = sqrt(z) can be expressed in polar coordinates
by the function f(z) = f(r*exp(i*theta)) = sqrt(r)*exp(i*theta/2) .4606Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 2.2 Transformations and Linear Mappings
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We now take our first look at the geometric interpretation of a complex function. If D is the domain of definition of the real-valued functions u(x, y) and v(x, y) , then the system of equations u = u(x, y) and v = v(x, y) describes a transformation or mapping from D in the xy-plane into the uv-plane. Therefore, the function f(z) = u(x, y)+i*v(x, y) can be considered as a mapping or transformation from the set D in the z-plane onto the range R in the w-plane.<img src="/view.aspx?si=4605//applications/images/app_image_blank_lg.jpg" alt="Section 2.2 Transformations and Linear Mappings" align="left"/>We now take our first look at the geometric interpretation of a complex function. If D is the domain of definition of the real-valued functions u(x, y) and v(x, y) , then the system of equations u = u(x, y) and v = v(x, y) describes a transformation or mapping from D in the xy-plane into the uv-plane. Therefore, the function f(z) = u(x, y)+i*v(x, y) can be considered as a mapping or transformation from the set D in the z-plane onto the range R in the w-plane.4605Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 2.1 Functions of a Complex Variable
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A complex valued function f of the complex variable z is a rule that assigns to each complex number z in a set D one and only one complex number w . We write w = f(z) and call w the image of z under f . The set D is called the domain of f , and the set of all images {w = f(z), z*epsilon*D} is called the range of f . As we saw in section 1.6, the term domain is also used to indicate a connected open set. When speaking about the domain of a function, however, mathematicians mean only the set of points on which the function is defined. This is a distinction worth noting.<img src="/view.aspx?si=4604//applications/images/app_image_blank_lg.jpg" alt="Section 2.1 Functions of a Complex Variable" align="left"/>A complex valued function f of the complex variable z is a rule that assigns to each complex number z in a set D one and only one complex number w . We write w = f(z) and call w the image of z under f . The set D is called the domain of f , and the set of all images {w = f(z), z*epsilon*D} is called the range of f . As we saw in section 1.6, the term domain is also used to indicate a connected open set. When speaking about the domain of a function, however, mathematicians mean only the set of points on which the function is defined. This is a distinction worth noting.4604Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 1.6 The Topology of Complex Numbers
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In this section we investigate some basic ideas concerning sets of points in the plane.<img src="/view.aspx?si=4603//applications/images/app_image_blank_lg.jpg" alt="Section 1.6 The Topology of Complex Numbers" align="left"/>In this section we investigate some basic ideas concerning sets of points in the plane.4603Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 1.5 The Algebra of Complex Numbers, Revisited
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The real numbers are deficient in the sense that not all algebraic operations on them produce real numbers. Thus, for sqrt(-1) to make sense, we must lift our sights to the domain of complex numbers. Do complex numbers have this same deficiency? That is, if we are to make sense out of expressions like sqrt(1+i) , must we appeal to yet another new number system? The answer to this question is no. It turns out that any reasonable algebraic operation we perform on complex numbers gives us complex numbers. In this respect, we say that the complex numbers are complete. Later we will learn how to evaluate intriguing algebraic expressions such as (-1)^i . For now we will be content to study integral powers and roots of complex numbers.<img src="/view.aspx?si=4602//applications/images/app_image_blank_lg.jpg" alt="Section 1.5 The Algebra of Complex Numbers, Revisited" align="left"/>The real numbers are deficient in the sense that not all algebraic operations on them produce real numbers. Thus, for sqrt(-1) to make sense, we must lift our sights to the domain of complex numbers. Do complex numbers have this same deficiency? That is, if we are to make sense out of expressions like sqrt(1+i) , must we appeal to yet another new number system? The answer to this question is no. It turns out that any reasonable algebraic operation we perform on complex numbers gives us complex numbers. In this respect, we say that the complex numbers are complete. Later we will learn how to evaluate intriguing algebraic expressions such as (-1)^i . For now we will be content to study integral powers and roots of complex numbers.4602Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John MathewsSection 1.4 The Geometry of Complex Numbers, Continued
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In Section 1.3 we saw that a complex number z = x+i*y could be viewed as a vector in the xy-plane whose tail is at the origin and whose head is at the point (x,y). A vector can be uniquely specified by giving its magnitude (i.e., its length) and direction (i.e., the angle it makes with the positive x-axis). In this section, we focus on these two geometric aspects of complex numbers.<img src="/view.aspx?si=4601//applications/images/app_image_blank_lg.jpg" alt="Section 1.4 The Geometry of Complex Numbers, Continued" align="left"/>In Section 1.3 we saw that a complex number z = x+i*y could be viewed as a vector in the xy-plane whose tail is at the origin and whose head is at the point (x,y). A vector can be uniquely specified by giving its magnitude (i.e., its length) and direction (i.e., the angle it makes with the positive x-axis). In this section, we focus on these two geometric aspects of complex numbers.4601Wed, 01 Oct 2003 00:00:00 ZDr. John MathewsDr. John Mathews