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18.075

Methods for Scientists and Engineers

3-0-9 units · Undergraduate

Covers functions of a complex variable; calculus of residues. Includes ordinary differential equations; Bessel and Legendre functions; Sturm-Liouville theory; partial differential equations; heat equation; and wave equations.

Fall 2004 · Advanced Calculus for Engineers

Topics

  • Number Systems and Algebra of Complex Numbers
  • Elementary Complex Functions, Part 1
  • Elementary Complex Functions, Part 2
  • Branch Points and Branch Cuts
  • Analytic Functions
  • Complex Integrals
  • Cauchy’s Formula, Properties of Analytic Functions
  • Taylor Series, Laurent Series
  • Laurent Series (cont.)
  • Properties of Laurent Series, Singularities
  • Singularities (cont.)
  • Residue Theorem
  • Evaluation of Real Definite Integrals, Case I
  • Evaluation of Real Definite Integrals, Case II
  • Evaluation of Real Definite Integrals, Case III
  • Evaluation of Real Definite Integrals, Case IV
  • Theorems for Contour Integration
  • Series and Convergence
  • Ordinary Differential Equations
  • Singular Points of Linear Second-Order ODEs
  • Frobenius Method
  • Frobenius Method - Examples
  • Frobenius Method (cont.) and a “particular type” of ODE
  • Bessel Functions
  • Properties of Bessel Functions
  • Modified Bessel Functions
  • Differential Equations Satisfied by Bessel Functions
  • Introduction to Boundary-Value Problems
  • Eigenvalues, Eigenfunctions, Orthogonality of Eigenfunctions
  • Boundary Value Problems for Nonhomogeneous PDEs
  • Sturm-Liouville Problem
  • Fourier Series
  • Fourier Sine and Cosine Series
  • Complete Fourier Series
  • Review of Boundary Value Problems for Nonhomogeneous PDEs

Readings

  • These chapters come from the text: Hildebrand, Francis. Advanced Calculus for Applications. 2nd ed. Englewood Cliffs: Prentice Hall, March 31, 1976. ISBN: 0130111899.
    Listed reading
  • Hildebrand, Francis. Advanced Calculus for Applications. 2nd ed. Englewood Cliffs: Prentice Hall, March 31, 1976. ISBN: 0130111899.
    Listed reading

Prerequisites & course information

Subject meets with 18.0751 Prereq: Calculus II (GIR) and 18.03 U (Spring) Not offered regularly; consult department 3-0-9 units Credit cannot also be received for 18.04

Sources