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### Integration with variables Notes_Part_2

Course: MATH 5587, Fall 2010
School: UCF
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Word Count: 838

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Re 1 z Figure 7.1. 1 Im z 1 Real and Imaginary Parts of f (z) = z . Therefore, if f (z) is any complex function, we can write it as a complex combination f (z) = f (x + i y) = u(x, y) + i v(x, y), of two inter-related real harmonic functions: u(x, y) = Re f (z) and v(x, y) = Im f (z). Before delving into the many remarkable properties of complex functions, let us look at some of the most basic examples. In each...

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Re 1 z Figure 7.1. 1 Im z 1 Real and Imaginary Parts of f (z) = z . Therefore, if f (z) is any complex function, we can write it as a complex combination f (z) = f (x + i y) = u(x, y) + i v(x, y), of two inter-related real harmonic functions: u(x, y) = Re f (z) and v(x, y) = Im f (z). Before delving into the many remarkable properties of complex functions, let us look at some of the most basic examples. In each case, the reader can directly check that the harmonic functions provided by the real and imaginary parts of the complex function are indeed solutions to the Laplace equation. Examples of Complex Functions (a) Harmonic Polynomials: As noted above, any complex polynomial is a linear combination, as in (7.2), of the basic complex monomials z n = (x + i y)n = un (x, y) + i vn (x, y). (7.7) Their real and imaginary parts, un , vn , are the harmonic polynomials that we previously constructed by applying separation of variables to the polar coordinate form of the Laplace equation (4.94). The general formula can be found in (4.110). (b) Rational Functions: Ratios p(z) f (z) = (7.8) q(z) of complex polynomials provide a large variety of harmonic functions. The simplest case is x y 1 = 2 -i 2 . (7.9) 2 z x +y x + y2 Its real and imaginary parts are graphed in Figure 7.1. Note that these functions have an interesting singularity at the origin x = y = 0, but are harmonic everywhere else. 1/19/12 219 c 2012 Peter J. Olver Figure 7.2. Real and Imaginary Parts of ez . A slightly more complicated example is the function f (z) = z-1 . z+1 (7.10) To write out (7.10) in real form, we multiply and divide by the complex conjugate of the denominator, leading to z-1 (z - 1)( z + 1) | z |2 + z - z - 1 x2 + y 2 - 1 2y = = = +i . 2 2 + y2 z+1 (z + 1)( z + 1) |z +1| (x + 1) (x + 1)2 + y 2 (7.11) Again, the real and imaginary parts are both harmonic functions away from the singularity x = -1, y = 0. Incidentally, the preceding manipulation can always be used to find the real and imaginary parts of general rational functions. (c) Complex Exponentials: Euler's formula f (z) = ez = ex cos y + i ex sin y (7.12) for the complex exponential yields two important harmonic functions: ex cos y and ex sin y, which are graphed in Figure 7.2. More generally, writing out ec z for a complex constant c = a + i b produces the complex exponential function ec z = ea x-b y cos(b x + a y) + i ea x-b y sin(b x + a y), (7.13) whose real imaginary and parts are harmonic functions for arbitrary a, b R. Some of these were found by applying the separation of variables method in Cartesian coordinates. (d) Complex Trigonometric Functions: The complex trigonometric functions are defined in terms of the complex exponential by adapting our earlier formulae (3.60): cos z = e i z + e- i z = cos x cosh y - i sin x sinh y, 2 e i z - e- i z = sin x cosh y + i cos x sinh y. sin z = 2i 220 c 2012 (7.14) 1/19/12 Peter J. Olver Re (log z) = log | z | Figure 7.3. Im (log z) = ph z Real and Imaginary Parts of log z. The resulting harmonic functions are products of trigonometric and hyperbolic functions, and can all be written as linear combinations of the harmonic functions (7.13) derived from the complex exponential. Note that when z = x is real, so y = 0, these functions reduce to the usual real trigonometric functions cos x and sin x. (e) Complex Logarithm: In a similar fashion, the complex logarithm log z is a complex extension of the usual real natural (i.e., base e) logarithm. In terms of polar coordinates z = r e i , the complex logarithm has the form log z = log(r e i ) = log r + log e i = log r + i . Thus, the logarithm of a complex number has real part Re (log z) = log r = log | z | = 1 2 (7.15) log(x2 + y 2 ), which is a well-defined harmonic function on all of R 2 save for a logarithmic singularity at the origin x = y = 0. It is, up to multiple, the logarithmic potential (6.104) corresponding to a delta function forcing concentrated at the origin -- which played a key role in our construction of the Green's function for the Poisson equation. The imaginary part Im (log z) = = ph z of the complex logarithm is the phase (or argument) of z, also not defined at the origin x = y = 0. Moreover, the phase is a multiply-valued harmonic function elsewhere, since it is only specified up to integer multiples of 2 . Each nonzero complex number z = 0 has an infinite number of possible values for its phase, and hence an infinite number of possible complex logarithms log z, differing from each other by an integer multiple of 2 i , which reflects the fact that e2 i = 1. In particular, if z = x > 0 is real and positive, then log z = log x agrees with the real logarithm, provided we choose ph x = 0. Alternative choices append some integer multiple of 2 i , and so ordinary real, positive numbers x > 0 1/19/12 221 c 2012 Peter J. Olver
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UCF - MATH - 5587
Figure 7.4.Real and Imaginary Parts ofz.also have complex logarithms! On the other hand, if z = x &lt; 0 is real and negative, then log z = log | x | + (2 k + 1) i is complex no matter which value of ph z is chosen. (This explains why one avoids defining
UCF - MATH - 5587
The proof of the converse - that any function whose real and imaginary components satisfy the CauchyRiemann equations is differentiable - will be omitted, but can be found in any basic text on complex analysis, e.g., [3, 65, 118]. Remark : It is worth poi
UCF - MATH - 5587
is analytic everywhere except for singularities at the points z = 3 and z = -1, where its denominator vanishes. Since f (z) = h1 (z) , z-3 where h1 (z) = ez (z + 1)21 is analytic at z = 3 and h1 (3) = 16 e3 = 0, we conclude that z = 3 is a simple (order
UCF - MATH - 5587
if and only if it has vanishing divergence: v = u v + = 0. x y (7.36)Incompressibility means that the fluid volume does not change as it flows. Most liquids, including water, are, for all practical purposes, incompressible. On the other hand, the flow is
UCF - MATH - 5587
Using formula (7.19) for the complex derivative, d = -i = u - i v, dz x y so = u, x = v. yThus, = v, and hence the real part (x, y) of the complex function (z) defines a velocity potential for the fluid flow. For this reason, the anti-derivative (z) is k
UCF - MATH - 5587
gDFigure 7.14.Mapping to the Unit Disk.Remark : In this section, we have focused on the fluid mechanical roles of a harmonic function and its conjugate. An analogous interpretation applies when (x, y) represents an electromagnetic potential function;
UCF - MATH - 5587
Figure 7.16.The Effect of = z 2 on Various Domains.obtained by cutting the complex plane along the negative real axis. On the other hand, vertical lines Re z = a are mapped to circles | | = ea . Thus, a vertical strip a &lt; Re z &lt; b is mapped to an annulu
UCF - MATH - 5587
zph zFigure 7.18.Complex Curve and Tangent.notation x(t) = ( x(t), y(t) ) to complex notation z(t) = x(t)+ i y(t). All the usual vectorial curve terminology - closed, simple (non-self intersecting), piecewise smooth, etc. - is employed without modific
UCF - MATH - 5587
Center: .1 Radius: .5Center: .2 + i Radius: 1Center: 1 + i Radius: 1Center: -2 + 3 i Radius: 3 2 4.2426Center: .2 + i Radius: 1.2806 Figure 7.21.Center: .1 + .3 i Radius: .9487Center: .1 + .1 i Radius: 1.1045Center: -.2 + .1 i Radius: 1.2042Airfoi
UCF - MATH - 5587
Example 7.35. The goal of this example is to construct an conformal map that takes a half disk D+ = | z | &lt; 1, Im z &gt; 0 (7.73) to the full unit disk D = cfw_ | | &lt; 1 . The answer is not = z 2 because the image of D+ omits the positive real axis, resulting
UCF - MATH - 5587
7.5. Applications of Conformal Mapping.Let us now apply what we have learned about analytic/conformal maps. We begin with boundary value problems for the Laplace equation, and then present some applications in fluid mechanics. We conclude by discussing h
UCF - MATH - 5587
Figure 7.25.A NonCoaxial Cable.Example 7.39. A non-coaxial cable. The goal of this example is to determine the electrostatic potential inside a non-coaxial cylindrical cable, as illustrated in Figure 7.25, with prescribed constant potential values on th
UCF - MATH - 5587
0 Figure 7.29.15 Fluid Flow Past a Tilted Plate.30Note that = ( 1, 0 ), and hence this flow satisfies the Neumann boundary conditions (7.95) on the horizontal segment D = . The corresponding complex potential is (z) = z, with complex velocity f (z) = (
UCF - MATH - 5587
on the unit disk D for an impulse concentrated at the origin; see Section 6.3 for details. How do we obtain the corresponding solution when the unit impulse is concentrated at another point = + i D instead of the origin? According to Example 7.25, the lin
UCF - MATH - 5587
as long as n = -1. Therefore, we can use the Fundamental Theorem of Calculus (which works equally well for real integrals of complex-valued functions), to evaluate n+1 1 -1 = n = 2 k + 1 odd, 0, 2 t + i (t - 1) 2 z n dz = = , n = 2 k even. n+1 P t = -1 n+
UCF - MATH - 5587
Figure 7.32.Orientation of Domain Boundary.Theorem 7.48. If f (z) is analytic on a bounded domain C, then f (z) dz = 0.(7.118)Proof : If we apply Green's Theorem to the two real line integrals in (7.109), we find u dx - v dy = - u v - x y = 0,v dx +
UCF - MATH - 5587
Proof : Note that the integrand f (z) = 1/(z - a) is analytic everywhere except at z = a, where it has a simple pole. If a is outside C, then Cauchy's Theorem 7.48 applies, and the integral is zero. On the other hand, if a is inside C, then Proposition 7.
UCF - MATH - 5587
0 Figure 7.36.15 Kutta Flow Past a Tilted Airfoil.30which remains asymptotically 1 at large distances. By Cauchy's Theorem 7.48 coupled with formula (7.123), if C is a curve going once around the disk in a counter-clockwise direction, then i 1 dz = - 2
UCF - MATH - 5587
is analytic in the disk | z | 2 since its only singularity, at z = 3, lies outside the contour C. Therefore, by Cauchy's formula (7.135), we immediately obtain the integral ez dz = z2 - 2 z - 3 f (z) i dz = 2 i f (-1) = - . z+1 2eCCNote: Path independe
UCF - MATH - 5587
Chapter 12 Dynamics of Planar MediaIn previous chapters we studied the equilibrium configurations of planar media - plates and membranes - governed by the two-dimensional Laplace and Poisson equations. In this chapter, we analyze their dynamics, modeled
UCF - MATH - 5587
In this manner, we arrive at the basic conservation law relating the heat energy density and the heat flux vector w. As in our one-dimensional model, cf. (4.3), the heat energy density (t, x, y) is proportional to the temperature, so (t, x, y) = (x, y) u(
UCF - MATH - 5587
for the diffusion equation. See [35; p. 369] for a precise statement and proof of the general theorem. Qualitative Properties Before tackling examples in which we are able to construct explicit formulae for the eigenfunctions and eigenvalues, let us see w
UCF - MATH - 5587
Theorem 12.1. Suppose u(t, x, y) is a solution to the forced heat equation ut = u + F (t, x, y), for (x, y) , 0 &lt; t &lt; c,where is a bounded domain, and &gt; 0. Suppose F (t, x, y) 0 for all (x, y) and 0 t c. Then the global maximum of u on the set cfw_ (t, x
UCF - MATH - 5587
so there are no non-separable eigenfunctions . As a consequence, the general solution to the initial-boundary value problem can be expressed as a linear combination u(t, x, y) =m,n = 1cm,n um,n (t, x, y) =m,n = 1cm,n e- m,n t vm,n (x, y)(12.41)of
UCF - MATH - 5587
Let us start with the equation for q(). The second boundary condition in (12.50) requires that q() be 2 periodic. Therefore, the required solutions are the elementary trigonometric functions q() = cos m or sin m , where = m2 , (12.53)with m = 0, 1, 2, .
UCF - MATH - 5587
15 10 5 -4 -2 -5 -10 -15 2 4Figure 12.3.The Gamma Function.Thus, at integer values of x, the gamma function agrees with the elementary factorial. A few other values can be computed exactly. One important case is when x = 1 . Using 2 the substitution t
UCF - MATH - 5587
Remark : The definition of a singular point assumes that the other coefficients do not both vanish there, i.e., either q(x0 ) = 0 or r(x0 ) = 0. If all three functions happen to vanish at x0 , we can cancel any common factor (x - x0 )k , and hence, withou
UCF - MATH - 5587
we find that the only non-zero coefficients un are when n = 3 k +1. The recurrence relation u3 k+1 = u3 k-2 (3 k + 1)(3 k) yields u3 k+1 = 1 . (3 k + 1)(3 k)(3 k - 2)(3 k - 3) 7 6 4 3The resulting solution isx3 k+1 . (3 k + 1)(3 k)(3 k - 2)(3 k - 3) 7 6
UCF - MATH - 5587
two different Frobenius expansions. Usually, this expectation is valid, but there is an important exception, which occurs when the indices differ by an integer. The general result is summarized in the following list: (i ) If r2 - r1 is not an integer, the
UCF - MATH - 5587
We have thus found the series solution (-1)k xm+2k . 22k k(k - 1) 3 2 (r + k)(r + k - 1) (r + 2)(r + 1) k=0 k=0 (12.93) So far, we not paid attention to the precise values of the indices r = m. In order to continue the recurrence, we need to ensure that t
UCF - MATH - 5587
where h0 = 0, while = limkhk = 1 +1 1 1 + + + , 2 3 k (12.102)hk - log k .5772156649 . . .is known as Euler's constant. All Bessel functions of the second kind have a singularity at the origin x = 0; indeed, by inspection of (12.101), we find that th
UCF - MATH - 5587
of the Bessel boundary value problem (12.5455) are the squares of the roots of the Bessel function of order m. The corresponding eigenfunctions are wm,n (r) = Jm (m,n r) , n = 1, 2, 3, . . . , m = 0, 1, 2, . . . , (12.112)defined for 0 r 1. Combining (12
UCF - MATH - 5587
t=0t = .04t = .08t = .12 Figure 12.6.t = .16 Heat Diffusion in a Disk.t = .212.5. The Fundamental Solution of the Heat Equation.As we learned in Section 4.1, the fundamental solution to the heat equation measures the temperature distribution result
UCF - MATH - 5587
for the planar heat equation is given by the linear superposition formula u(t, x, y) = 1 4 t f (, ) e- [ (x-)2+(y-)2 ]/(4 t)d d.(12.125)We can interpret the solution formula (12.125) as a two-dimensional convolution u(t, x, y) = F (t, x, y) f (x, y)
UCF - MATH - 5587
Vibration of a Rectangular Drum Let us first consider the vibrations of a membrane in the shape of a rectangle R= 0 &lt; x &lt; a, 0 &lt; y &lt; b ,with side lengths a and b, whose edges are fixed to the (x, y)plane. Thus, we seek to solve the wave equation utt = c2
UCF - MATH - 5587
A table of their values (for the case c = 1) can be found in the preceding section. The Bessel roots do not follow any easily discernible pattern, and are not rational multiples of each other. This result, known as Bourget's hypothesis, [142; p. 484], was
UCF - MATH - 5587
following table, we display a list of all relative vibrational frequencies (12.158) that are &lt; 6. Once the lowest frequency 0,1 has been determined - either theoretically, numerically or experimentally - all the higher overtones m,n = m,n 0,1 are simply o
UCF - MATH - 5587
For example, on a unit square R = 0 &lt; x, y &lt; 1 , an accidental degeneracy occurs whenever m2 + n2 = k 2 + l2 (12.163) for distinct pairs of positive integers (m, n) = (k, l). The simplest possibility arises whenever m = n, in which case we can merely reve
UCF - MATH - 5587
Chapter 9 Linear and Nonlinear Evolution EquationsIn this chapter, we analyze several of the most important evolution equations, both linear and nonlinear, involving a single spatial variable. Our first stop is to revisit the heat equation. We introduce
UCF - MATH - 5587
Chapter 3 Fourier SeriesJust before 1800, the French mathematician/physicist/engineer Jean Baptiste Joseph Fourier made an astonishing discovery. Through his deep analytical investigations into the partial differential equations modeling heat propagation
UCF - MATH - 5587
Chapter 8 Fourier TransformsFourier series and their ilk are designed to solve boundary value problems on bounded intervals. The extension of Fourier methods to the entire real line leads naturally to the Fourier transform, an extremely powerful mathemat
UCF - MATH - 5587
Chapter 6 Generalized Functions and Green's FunctionsBoundary value problems, involving both ordinary and partial differential equations, can be profitably viewed as the infinite-dimensional function space versions of finite dimensional systems of linear
UCF - MATH - 5587
Math 5587 September 8, 2011Homework #1Problems: Chapter 1: 1.1ae, 1.2b,d, 1.5a,e, 1.6, 1.12a, 1.16ad, 1.18, 1.19, 1.20, 1.24. Chapter 2: 2.1 2, 3c,e, 4, 6.Due: Thursday, September 15
UCF - MATH - 5587
Math 5587 September 20, 2011Homework #2Problems: Chapter 2: 2.2 2.3 2a, 3b, 9, 17, 26, 27. 2, 5, 14, 15.Due: Thursday, September 29 First Midterm: Tuesday, October 11 Will cover chapters 1 &amp; 2. You will be allowed to use one 8&quot; 11&quot; sheet of notes. Note
UCF - MATH - 5587
Math 5587 September 29, 2011Homework #3Problems: Chapter 2: 2.4 2, 3, 4c,d, 8, 11, 12.Also, in 2.4.8, determine the domain of influence of the point (0,2) and the domain of dependence of the point (3,-1). Discuss what these tell you about the solution.
UCF - MATH - 5587
Math 5587 October 13, 2011Homework #4Problems: Chapter 3: 3.1 3.2 2b, 5. 1, 2g, 3a, 5, 6a,g, 15a,d, 16a,d, 24, 25, 34, 35, 41b, 52, 53.Due: Thursday, October 20
UCF - MATH - 5587
Math 5587 October 25, 2011Homework #5Problems: Chapter 3: 3.3 1, 2, 8. 3.4 2b, 3c, 7, 9 (just use one of the two methods). 3.5 2b,c,d, 4, 8, 11a,b,c. Due: Tuesday, November 1 Second Midterm: Thursday, November 17 Will cover chapters 3 &amp; 4. You will be a
UCF - MATH - 5587
Math 5587 November 3, 2011Homework #6Problems: Chapter 3: 3.5 13, 21c,e, 22b,c, 27b,d, 30, 31, 35a, 42. Chapter 4: 4.1 2, 4c, 10, 17a,b. Due: Thursday, November 10 Second Midterm: Thursday, November 17 Will cover chapters 3 &amp; 4. You will be allowed to u
UCF - MATH - 5587
Math 5587 November 10, 2011Homework #7Problems: Chapter 4: 4.2 3a, 4b,e, 8, 14a,d,e, 26. 4.3 4, 7, 10c, 11, 12a, 16, 24a, 29, 31. Due: Tuesday, November 22 Second Midterm: Thursday, November 17 Will cover chapters 3 &amp; 4. You will be allowed to use one 8
UCF - MATH - 5587
Math 5587 December 6, 2011Homework #8Problems: Chapter 4: 4.4 2a,e,f, 12a,e,f, 13, 17a,b. Chapter 6: 6.1 1b,d, 2d, 3, 5b, 8, 13, 19, 35. 6.2 2, 6. 6.3 1, 2, 6. Due: Tuesday, December 13 Final Exam: Take Home, to be handed out on Tuesday, December 13 and
UCF - MATH - 5587
Chapter 2 Linear and Nonlinear WavesOur exploration of the vast mathematical continent that is partial differential equations will begin with simple first order equations. In applications, first order partial differential equations are most commonly used
UCF - MATH - 5587
Chapter 5 Numerical Methods: Finite DifferencesAs you know, the differential equations that can be solved by an explicit analytic formula are few and far between. Consequently, the development of accurate numerical approximation schemes is essential for
UCF - MATH - 5587
Chapter 11 Numerical Methods: Finite ElementsIn Chapter 5, we introduced the first, the oldest, and in many ways the simplest class of numerical algorithms for approximating the solutions to partial differential equations: finite differences. In the pres
UCF - MATH - 5587
Chapter 10 A General Framework for Linear Partial Differential EquationsBefore pressing on to the higher dimensional forms of the heat, wave, and Laplace/ Poisson equations, it is worth taking some time to develop a general, abstract, linear algebraic fr
UCF - MATH - 5587
Chapter 12 Partial Differential Equations in SpaceAt last we have reached the ultimate rung of the dimensional ladder (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. As in the one- and
UCF - MATH - 5587
Chapter 4 Separation of VariablesThere are three paradigmatic linear second order partial differential equations that have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes v
UCF - MATH - 5587
Chapter 1 What are Partial Differential Equations?Let us begin by specifying our object of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d4 u du + u2 = c
University of Florida - CEG - 4012
Lecture 1 Review of Geostatic Stresses Unit WeightsYw = unit weight of water Ym = moist unit weight of unsaturated soil Ysat = unit weight of saturated soil y' = &quot;effective&quot; unit weight of soil = (Ysat - Yw) if soil saturated= Ym if soil not saturated=
University of Florida - CEG - 4012
Geostatic Stresses These self-weight stresses (ay, a'y, ah' a'h) are called geostatic stresses For a level surface there are no shear forces induced by the geostatic stresses, and therefore they are also principal stresses: a, =ay and a3 = ah Karl Terzag
University of Florida - CEG - 4012
using the effective weight concept:= 0&quot; h6'(100 pet + 4'(118-62.4 pet + 6'(126-62.4 pet+4'(120-62.4 pet 230 pst: 1434 pst= 600 pst + 222 pst + 382 pst +=-K (J'y : 0.5 (1434 pst) : 717 pst 14' (62.4 pet): 874 pst U(a' h + u): 717 pst + 874 pst: 15
University of Florida - CEG - 4012
S1 Stresses Changes Due to Surface Loads (Aerv) o Stresses within a soil mass will change as a result of surface loads. The change in total stress spreads and diminishes with distance from the load. Equations and charts are available to calculate both th