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Course: BA 271, Fall 2008
School: Oregon State
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Coursehero >> Oregon >> Oregon State >> BA 271

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Oregon State - BA - 271
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Oregon State - BA - 271
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Oregon State - BA - 271
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Oregon State - BA - 271
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Oregon State - BA - 271
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Oregon State - BA - 271
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Oregon State - BA - 271
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Oregon State - BA - 271
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UGA - MATH - 3000
Math 3000: Linear AlgebraLecturer: Dr. Jason Parsley Office: Boyd GSRC 407 Office phone: 542-2562 Office hours: Tu. 3:30-4:30 pm, W. 11-12, and also by appointment Email: parsley@math.uga.edu Please do not email me via WebCT; I do not read it often.
U. Houston - ECE - 6382
ECE 6382Fall 2008Analytic ContinuationD. R. Wilton ECE Dept.Analytic Continuation of Functionsm It seems clear how to extend linear combinations of the elementary functions* $ 1, x 1 , x 2 , x 3 , x 4 ,x 5 ,L ,x n ,L x m12,Pn ( x ) (ra
U. Houston - ECE - 6382
ECE 6382Fall 2008Applications of Bessel FunctionsD. R. Wilton ECE Dept.z-Independent Circular Cylinder Modes Dirichlet Case Assume that satisifies the 2D wave equation :a ( a, ) = 0(2+ k 2 ) ( , ) = 0,k = 2 = v , v is the
U. Houston - ECE - 6382
ECE 6382Fall 2008Applications of Bessel FunctionsD. R. Wilton ECE Dept.z-Independent Circular Cylinder Modes Dirichlet CaseAssume that satisifies the 2D wave equation :a ( a, ) = 0(D2+ k 2 ) ( , ) = 0,k = 2 = v , v is the
U. Houston - ECE - 6382
ECE 6382Fall 2007Bessel FunctionsD. R. Wilton ECE Dept.Wave Equation in Cylindrical CoordinatesSource - free scalar wave equation in cylindrical coordinates ( , , z ) : 2 + k 2 = 0, 1 k = 2 / 1 2 2 + 2 + k 2 = 0 + 2 2 z As
U. Houston - ECE - 6382
ECE 6382Fall 2007 Prof. David R. Jackson ECE Dept.Branch Points and Branch CutsPreliminaryConsiderf (z) = zz1/ 21/ 2z = r e jj 1/ 2= (r e)= r e j / 2Choosez =1 = 0: = 2 : = 4 :z1/ 2 = 1 z1/ 2 = -1 z1/ 2 = 1There are t
U. Houston - ECE - 6382
ECE 6382Fall 2008Differentiation of Functions of a Complex VariableD. R. Wilton ECE Dept.Differentiation of Functions of a Complex Variable Function of a complex variable : z = x + iy, w = u + iv w = f ( z ) = u ( x, y ) + iv ( x, y ) ( e.g.,
U. Houston - ECE - 6382
ECE 6382Fall 2008Differentiation of Functions of a Complex VariableD. R. Wilton ECE Dept.Differentiation of Functions of a Complex VariableFunction of a complex variable : z = x + iy, w = u + iv w = f ( z ) = u ( x, y ) + iv ( x, y ) where u (
U. Houston - ECE - 6382
ECE 6382Fall 2008Integration in the Complex PlaneD. R. Wilton ECE Dept.Line Integrals in the Complex PlaneConsiderCy n on C between zn-1 and zn N b = zN n 1 z1 2 z2a = z03z3.zn.z N -1 Consider the sums I N = f ( n )(
U. Houston - ECE - 6382
ECE 6382Fall 2008Integration in the Complex PlaneD. R. Wilton ECE Dept.Line Integrals in the Complex PlaneConsiderC y n on C between zn-1 and znn3 . 2 z2 z3 1 z1a = z0zn. N b = zN z N -1Consider the sums IN =N n =1f ( n )
U. Houston - ECE - 6382
Name_0BTime returned _1BECE 63822BFinal Exam3BFall, 20084BYour exam solution should be returned no later than 5:00 PM, Friday, Dec. 19, 2008.5BI certify that all the work contained in this exam is my own. I have neither given nor re
U. Houston - ECE - 6382
Name_ Time received _ Time returned _ECE 6382 Final Exam Fall, 2006Your exam solution should be returned no later than 72 hours after you received it.I certify that all the work contained in this exam is my own. I have neither given nor receive
U. Houston - ECE - 6382
0BName_1BTime returned _2BECE 63823BFinal Exam4BFall, 20085BYour exam solution should be returned no later than 5:00 PM, Friday, Dec. 19, 2008.I certify that all the work contained in this exam is my own. I have neither given n
U. Houston - ECE - 6382
ECE 6382Fall 2008Introduction to Green's FunctionsD. R. Wilton ECE Dept.Static Potential of Point Sources It is well known that the free space static potential at an observation point r due to a point charge Q at r is (r ) = Q , 4 0 r - r
U. Houston - ECE - 6382
ECE 6382Fall 2008Introduction to Green's FunctionsD. R. Wilton ECE Dept.Static Potential of Point SourcesIt is well known that the free space static potential at an observation point r due to a point charge Q at r is Q (r ) = , 4 0 r - r On t
U. Houston - ECE - 6382
ECE 6382Fall 2007Green's Functions for SOLDEsD. R. Wilton ECE Dept.Green's Functions for General SOLDEs The most general operator of second order is d2 d L = a0 ( x) 2 + a1 ( x) + a2 ( x) dx dx The initial value problem satisfies the followi
U. Houston - ECE - 6382
ECE 6382Fall 2008Green's Functions for the Stretched String ProblemD. R. Wilton ECE Dept.Motivation: System Impulse ResponseN th order diff. eq.Lvout = vin+ vin -L+ vout -+ ( t) - (t )L+ h(t ) - h (t ) Impulse response : L
U. Houston - ECE - 6382
ECE 6382Fall 2008Green's Functions for the Stretched String ProblemD. R. Wilton ECE Dept.Motivation: System Impulse ResponseN th order diff. eq.} Lvout = vin+ vin -L+ vout - ( t) - ( t)+L+ h( t) -h( t)Impulse response :
U. Houston - ECE - 6382
ECE 6382Fall 2008Evaluation of Definite Integrals Via the Residue TheoremD. R. Wilton ECE Dept.Recall for real integrals,2Review of Cauchy Principal Value Integrals 1/x-0 dx 2 dx dx 0 2 = + = ln x x =-1 + ln x x =0 = I = 0 x -1 x -1 x
U. Houston - ECE - 6382
ECE 6382Fall 2008Evaluation of Definite Integrals Via the Residue TheoremD. R. Wilton ECE Dept.Recall for real integrals,2Review of Cauchy Principal Value Integrals 1/x-0 dx 2 dx dx 0 2 I = = + = ln x x =-1 + ln x x =0 = -1 x -1 x 0 x
U. Houston - ECE - 6382
ECE 6382Fall 2008Introduction to the Theory of Complex VariablesD. R. Wilton ECE Dept.Some Applications of Complex VariablesExtension of real to complex variables : e -ikx - phase factor for a traveling, lossless plane wave ( k real) k k- ik -
U. Houston - ECE - 6382
ECE 6382Fall 2008 Prof. David R. Jackson ECE Dept.Branch Points and Branch CutsPreliminaryConsiderf (z) = zz1/ 21/ 2z = r e jj 1/ 2= (r e)= r e j / 2Choosez = r =1 = 0: = 2 : = 4 :z1/ 2 = 1 z1/ 2 = -1 z1/ 2 = 1There a
U. Houston - ECE - 6382
ECE 6382Fall 2008 Prof. David R. Jackson ECE Dept.Branch Points and Branch CutsPreliminaryConsiderf ( z ) = z1/ 2 z1/ 2z = r e jj 1/ 2=(re)= rej / 2Choosez = r =1 = 0:z1/ 2 = 1 z1/ 2 = -1 z1/ 2 = 1 = 2 : = 4 :There a
U. Houston - ECE - 6382
ECE 6382Fall 2008Legendre FunctionsD. R. Wilton ECE Dept.Wave Equation in Spherical CoordinatesSource - free scalar wave equation in spherical coordinates (r , , ) : 2 + k 2 = 0, 1 2 r2 r r r k = 2 / 2 1 + k 2 = 0 + 2 2 2 r sin
U. Houston - ECE - 6382
ECE 6382Fall 2008Legendre FunctionsD. R. Wilton ECE Dept.Wave Equation in Spherical CoordinatesSource - free scalar wave equation in spherical coordinates (r , , ) : D 2 + k 2 = 0, k = 2 / 2 1 sin + 2 2 + k 2 = 0 2 r sin 1 < 2
U. Houston - ECE - 6382
ECE 6382Fall 2008Functions of a Complex Variable as MappingsD. R. Wilton ECE Dept.A Function of a Complex Variable as a Mapping A function of a complex variable, w = f ( z ) , is usually viewed as a mapping from the complex z to the complex w
U. Houston - ECE - 6382
ECE 6382Fall 2008Functions of a Complex Variable as MappingsD. R. Wilton ECE Dept.A Function of a Complex Variable as a MappingA function of a complex variable, w = f ( z ) , is usually viewed as a mapping from the complex z to the complex w p
U. Houston - ECE - 6382
ECE 6382Fall 2007Pole and Product Expansions, Series SummationD. R. Wilton ECE Dept.Pole Expansion of Meromorphic FunctionsMittag - Leffler1 Theorem : f ( z ) has simple poles at z = an , n = 1,. , where 0 < a1 < a2 < a3 < f ( z ) has residu
U. Houston - ECE - 6382
ECE 6382Fall 2007Pole and Product Expansions, Series SummationD. R. Wilton ECE Dept.Pole Expansion of Meromorphic FunctionsMittag - Leffler1 Theorem : f ( z ) has simple poles at z = an , n = 1,K , where 0 < a1 < a2 < a3 < L f ( z ) has residu
U. Houston - ECE - 6382
ECE 6382Fall 2008Power Series RepresentationsD. R. Wilton ECE Dept.Geometric Series Consider the sumConsiderNoting thatSN = 1+ z + z +2+ z = znN n =0Ny1z <1 z >1zS N = z + z 2 + 1 - z N +1 1- z+ z N +1 ,we have that S N -
U. Houston - ECE - 6382
ECE 6382Fall 2008Power Series RepresentationsD. R. Wilton ECE Dept.Geometric SeriesConsider the sum Noting thatConsiderSN = 1+ z + z2 +L + z N =N n=0zny1z <1 z >1zS N = z + z 2 + L + z N +1 , we have that S N - zS N = ( 1 - z ) S
U. Houston - ECE - 6382
ECE 6382Fall 2008The Residue Theorem and Residue EvaluationD. R. Wilton ECE Dept.The Residue TheoremCConsider a line integral about a path enclosing an isolated singular point:ey z0 x f ( z ) dzCExpand f(z) in a Laurent series, deform
U. Houston - ECE - 6382
ECE 6382Fall 2008The Residue Theorem and Residue EvaluationD. R. Wilton ECE Dept.The Residue TheoremConsider a line integral about a path enclosing an isolated singular point:ey z0Cxy f ( z ) dzCExpand f(z) in a Laurent series, defo
U. Houston - ECE - 6382
ECE 6382Fall 2008Scalar Products, NormsD. R. Wilton ECE Dept.Ordinary Vectors with Complex CoefficientsA x ^ ^ ^ A = Ax x + Ay y + Az z = ( Ax , Ay , Az ) = y A z A Scalar product : A* <B* * A, B > = Ax Bx + A* By + Az Bz yAx , Ay
U. Houston - ECE - 6382
ECE 6382Fall 2008Series Solutions of SOLDEs with Regular Singular PointsD. R. Wilton ECE Dept.Series Solutions of Second Order Linear Differential Equations (SOLDEs)EXAMPLE 1: Indicial equation with complex roots (Case 1) x 1 x 2 y + xy +
U. Houston - ECE - 6382
ECE 6382Fall 2008Series Solutions of SOLDEs with Regular Singular PointsD. R. Wilton ECE Dept.Series Solutions of Second Order Linear Differential Equations (SOLDEs)EXAMPLE 1: Indicial equation with complex roots (Case 1) x 2 2 x 2 yL+ xy + y
U. Houston - ECE - 6382
ECE 6382Fall 2008SingularitiesD. R. Wilton Adapted from notes of Prof. David R. Jackson ECE Dept.SingularityA point zs is a singularity of the function f(z) if the function is not analytic at zs.(The function does not necessarily have to go
U. Houston - ECE - 6382
ECE 6382Fall 2008SingularitiesD. R. Wilton Adapted from notes of Prof. David R. Jackson ECE Dept.SingularityA point zs is a singularity of the function f(z) if the function is not analytic at zs.(The function does not necessarily have to go
U. Houston - ECE - 6382
ECE 6382Fall 2008Second Order Linear Differential EquationsD. R. Wilton ECE Dept.Separation of VariablesConsider the scalar, 3D wave equation in spherical coordinates : where2 2 (r , , ) + k 2 (r , , ) = 02 1 6 2 1 1 = 2 r + 2 s
U. Houston - ECE - 6382
ECE 6382Fall 2008Spherical Bessel FunctionsD. R. Wilton ECE Dept.Spherical Bessel FunctionsSpherical Bessel differential equation : d2y dy x + 2 x + x 2 - n(n + 1) y = 0 dx 2 dx or2d 2 dy x + x 2 - n(n + 1) y = 0 (self - adjoint fo
U. Houston - ECE - 6382
ECE 6382Fall 2008Spherical Bessel FunctionsD. R. Wilton ECE Dept.Spherical Bessel FunctionsSpherical Bessel differential equation : d2y dy x + 2 x + 2 - n(n + 1) = 0 x y 2 dx dx or2d 2 dy 2 x + x y - n(n + 1) = 0 dx dx With the
U. Houston - ECE - 6382
ECE 6382Fall 2007 Prof. David R. Jackson ECE Dept.Steepest-Descent MethodSteepest Descent MethodComplex Integral:I ( ) = f ( z ) e g ( z ) dzCThe functions f(z) and g(z) are analytic, so that the path C may be deformed if necessary.Sad
U. Houston - ECE - 6382
ECE 6382Fall 2008 Prof. Donald R. Wilton ECE Dept.Based on Original Notes of Prof. David R. JacksonSteepest Descent MethodComplex Integral:I ( ) = f ( z ) e g ( z ) dzCThe functions f(z) and g(z) are analytic, so that the path C may be d
U. Houston - ECE - 6382
ECE 6382Fall 2008 Prof. Donald R. Wilton ECE Dept.Based on Original Notes of Prof. David R. JacksonSteepest Descent MethodComplex Integral:I ( ) =Cf ( z ) e g ( z ) dzThe functions f(z) and g(z) are analytic, so that the path C may be de
U. Houston - ECE - 6382
ECE 6382Fall 2008Sturm-Liouville TheoryD. R. Wilton ECE Dept.Sturm-Liouville Differential Equations The Sturm - Liouville differential equation has the form d2y dy p0 ( x) 2 + p1 ( x) + ( p2 ( x) - ) y = f ( x) dx dx where the constant param
U. Houston - ECE - 6382
ECE 6382Fall 2008Sturm-Liouville TheoryD. R. Wilton ECE Dept.Sturm-Liouville Differential EquationsThe Sturm - Liouville differential equation has the form d2y dy p0 ( x) 2 y p1 ( x) + ( p2 ( x) - ) y = f ( x) + dx dx where the constant para
U. Houston - ECE - 6382
Note Title
U. Houston - ECE - 6382
Cauchy Principal Value Integrals:To evaluateI = -(dx , x + 1 ( x + 1)2)CR : z = R ei ,z =i-Rdz = iR ei dC :Rz + 1 = ei , dz = i ei dconsider the integralz = -i -1- R dz dz = lim + + + 2 z 2 + 1 ( z + 1) R - R -
U. Houston - ECE - 6382
Note Title
U. Houston - ECE - 6382
Note Title
U. Houston - ECE - 6382
ECE 6341Spring 2005 Prof. David R. Jackson ECE Dept.Notes 33Steepest Descent MethodComplex Integral:I () = f ( z ) eC g( z)dzThe functions f(z) and g(z) are analytic, so that the path C may be deformed if necessary.Saddle Point:g
Stanford - VVEGA - 500
Media Aggression Against WomenSyllabusYour final grade will be based on the following requirements: midterm exam, Tues., Jul. 18, (30%) 5 -6 pg paper due Thurs., Aug. 10, (20%) final exam, Thurs., Aug. 17, (40%) class participation (10%)Cour
Stanford - VVEGA - 500
Sexually Explicit Media Effects ResearchMedia Effects ParadigmSender Message / Medium Receiver & Media EffectsLimited Effects Active Audience Media has little effect in changing people's opinion or behavior. proponents of free speech Powerful