Math 241 Calculus, Part IV (4h. 1c.u.) Text: Zill, Dennis and Cullen ...

Math 241
Calculus, Part IV (4h. 1c.u.)
Fourier series and partial orthogonal functions; Fourier series; separable partial
differential equations: differential equations; solutions of the wave, heat and Laplace equations;
boundary value problems in Cartesian and other coordinate systems
Introduction to complex analysis: functions of a complex variable; complex integration; series and residue
theory
Use of symbolic manipulation and graphics software.
Text: Zill, Dennis and Cullen, Michael: Advanced Engineering Mathematics, Second Edition
(Sudbury, MA: Jones & Bartlett Publishers, ©1999)
Syllabus:
Core Maple
Chapter Section & Topic Problems Problems
Fourier Series and Partial Differential Equations 647
12 Orthogonal Functions and Fourier Series 651
12.1 Orthogonal Functions 652 3, 9, 11, 15
12.2 Fourier Series 656 5, 13, 15, 17
12.3 Fourier Cosine and Sine Series 662 7, 11, 15, 29, 35, 39 43
12.4 Complex Fourier Series and Frequency Spectrum 669 3, 8, 9
12.5 Sturm-Liouville Problem 672 1, 3, 7 9
12.6 Bessel and Legendre Series 678 3, 7, 15, 19 11
13 Boundary-Value Problems in Rectangular Coordinates 687
13.1 Separable Partial Differential Equations 688 5, 7, 9, 17, 23
13.2 Classical Equations and Boundary-Value Problems 692 1, 5, 9
13.3 Heat Equation 697 1, 5 8
13.4 Wave Equation 700 3, 7, 11 21
13.5 Laplace's Equation 704 5, 9, 15 19
13.6 Nonhomogeneous Equations and Boundary Conditions 709 1, 5, 11
13.7 Orthogonal Series Expansions 711 3, 5, 7 9
13.8 Fourier Series in Two Variables 715 1, 3, 5
14 Boundary-Value Problems in Other Coordinate Systems 719
14.1 Problems Involving Laplace's Equation in Polar Coordinates 720 3, 5, 11 15
14.2 Problems in Polar and Cylindrical Coordinates 724 2, 5, 9 13
14.3 Problems in Spherical Coordinates 730 1, 5, 9
15 Integral Transform Method 735
15.1 Error Function 736 1, 7
15.2 Applications of the Laplace Transform 737 1, 3, 7, 15, 21, 27 11, 29
15.3 Fourier Integral 745 3, 9, 15, 19
15.4 Fourier Transforms 750 1, 7, 11, 15, 21 5
15.5 Fast Fourier Transform 756 (Optional) 1, 3, 7 9

Core Maple
Chapter Section & Topic Problems Problems
Complex Analysis 785
17 Functions of a Complex Variable 789
17.3 Sets of Points in the Complex Plane 798 3, 5, 11, 13, 17, 25
17.4 Functions of a Complex Variable; Analyticity 800 1, 5, 9, 17, 21, 23, 31, 33, 37, 43
17.5 Cauchy-Riemann Equations 806 1, 5, 11, 15, 20, 27
17.6 Exponential and Logarithmic Functions 811 3, 11, 15, 17, 25, 37, 45, 47
17.7 Trigonometric and Hyperbolic Functions 818 3, 9, 13, 17, 24
17.8 Inverse Trigonometric and Hyperbolic Functions 822 1, 5, 11, 13
18 Integration in the Complex Plane 827
18.1 Contour Integrals 828 3, 5, 13, 19, 23, 25, 29, 33
18.2 Cauchy-Goursat Theorem 833 3, 9, 13, 19, 21, 23
18.3 Independence of Path 838 2, 4, 9, 15, 19, 23
18.4 Cauchy's Integral Formula 845 3, 5, 11, 17, 21, 23
19 Series and Residues 851
19.1 Sequences and Series 852 3, 7, 11, 15, 19, 23, 27
19.2 Taylor Series 857 3, 11, 17, 21, 27, 31
19.3 Laurent Series 863 3, 9, 15, 21, 25, 27
19.4 Zeros and Poles 870 1, 5, 9, 13, 19, 23
19.5 Residues and Residue Theorem 873 3, 9, 13, 17, 21, 27, 31
19.6 Evaluation of Real Integrals 879 3, 7, 11, 17, 27, 31, 35
OLD EXAM QUESTIONS also form a part of the core.
The core problems indicate the kind of basic problems you will need to be able to solve by hand. They also provide a guide to
the basic level of difficulty to be expected on the final exam.
Note: All sections of Math 241 have a COMMON FINAL EXAM
The following sections contain material covered in previous courses. Students are encouraged to review this
material as needed.
17 Functions of a Complex Variable 789
17.1 Complex Numbers 790
17.2 Form of Complex Numbers; Powers and Roots 793