18.04
Complex Variables with Applications
Complex algebra and functions; analyticity; contour integration, Cauchy's theorem; singularities, Taylor and Laurent series; residues, evaluation of integrals; multivalued functions, potential theory in two dimensions; Fourier analysis, Laplace transforms, and partial differential equations.
Spring 2018 · Complex Variables with Applications
Topics
- Topic 1: Complex Algebra and the Complex Plane
- Topic 1: Complex Algebra and the Complex Plane (cont.)
- Topic 2: Analytic Functions
- Topic 2: Analytic Functions (cont.)
- Review of 18.02 Multivariable Calculus
- Topic 3: Cauchy’s Theorem
- Topic 3: Cauchy’s Theorem (cont.)
- Topic 4: Cauchy Integral Formula
- Topic 4: Cauchy Integral Formula (cont.)
- Topic 5: Harmonic Functions
- Topic 5: Harmonic Functions (cont.); Topic 6: Two Dimensional Hydrodynamics and Complex Potentials
- Topic 6: Two Dimensional Hydrodynamics and Complex Potentials (cont.)
- Topic 7: Taylor and Laurent Series
- Topic 7: Taylor and Laurent Series (cont.)
- Topic 8: Residue Theorem
- Topic 8: Residue Theorem (cont.)
- Topic 9: Definite Integrals Using the Residue Theorem
- Topic 9: Definite Integrals Using the Residue Theorem (cont.)
- Topic 10: Conformal Transformations
- Topic 10: Conformal Transformations (cont.)
- Topic 11: Argument Principle
- Topic 11: Argument Principle (cont.)
- Topic 12: Laplace Transform
- Topic 12: Laplace Transform (cont.)
- Topic 13: Analytic Continuation and the Gamma Function
Readings
- There is no required textbook. We will use the following written resources:
- Brown, James Ward and Ruel V. Churchill. Complex Variables and Applications. 9th ed. McGraw-Hill Education, 2013. ISBN: 9780073383170.
Fall 1999
Topics
- 1.1 Complex Algebra. Complex Plane. Motivation and History
- 1.2 Polar Form. Complex Exponential. DeMoivre’s Theorem
- 1.3 Newton’s Method. Fractals
- 2.1 Complex Functions. Analyticity
- 2.2 Cauchy-Riemann Eqns. Harmonic Functions
- 2.3 Exponential and Trig. Functions. Logarithmic Function
- 2.4 Branch Cuts. Applications
- 2.5 Complex Powers and Inverse Trig. Functions
- 3.1 Contour Integrals
- 3.2 Path Independence
- 3.3 Cauchy’s Theorem
- 3.4 Cauchy’s Integral Formula
- 3.5 Liuville’s Theorem. Mean Value and Max. Modulus
- 3.6 Dirichlet Problem
- 4.1 Taylor Series. Radius of Convergence
- 4.2 Laurent Series
- 4.3 Zeros. Singularities. Point at Infinity
- 5.1 Residue Theorem. Integrals over the Unit Circle
- 5.2 Real Integrals. Conversion to Complex Contours
- 5.3 Trig. Integrals. Jordan’s Lemma
- 5.4 Indented Contours. Principal Value
- 5.5 Integrals Involving Multi-valued Functions
- 5.6 Argument Principle and Rouche’s Theorem
- 6.1 Complex Fourier Series
- 6.2 Oscillating Systems. Periodic Functions
- 6.3 Applications of Fourier Series
- 6.4 Fourier Transform and Applications
- 6.5 Laplace Transform and Inversion Formula
- 7.1 Invariance of Laplace’s Eqn. Conformality
- 7.2 Inversion Mapping. Bilinear Mappings
- 7.3 Examples and Applications
- 7.4 More Examples (if time permits)
Readings
- All readings are from the required textbook:
- Saff, Edward and Arthur David Snider. Fundamentals of Complex Analysis. 3nd ed. Upper Saddle River: Prentice Hall, 1993. ISBN: 0133274616.
- Week #: 1 · Lec #: 1 · Readings: Sections 1.1 & 1.2
- Lec #: 2 · Readings: Sections 1.3 - 1.5
- Week #: 2-3 · Lec #: 3 · Readings: Handout
- Lec #: 4 · Readings: Sections 2.1 - 2.3
- Lec #: 5 · Readings: Sections 2.4 - 2.6
- Lec #: 6 · Readings: Sections 3.1 & 3.2
- Lec #: 7 · Readings: Handout
- Lec #: 8 · Readings: Section 3.3
- Week #: 4-5 · Lec #: 9 · Readings: Sections 4.1 & 4.2
- Lec #: 10 · Readings: Sections 4.3 & 4.4a
- Lec #: 11 · Readings: Section 4.4b
- Lec #: 12 · Readings: Section 4.5
- Lec #: 13 · Readings: Section 4.6
- Week #: 6 · Lec #: 16 · Readings: Sections 5.1, 5.2 and handout
- Lec #: 17 · Readings: Section 5.5
- Lec #: 18 · Readings: Sections 5.6 & 5.7
- Week #: 7-8 · Lec #: 19 · Readings: Sections 6.1, 6.2 and handout
- Lec #: 20 · Readings: Section 6.3
- Lec #: 21 · Readings: Section 6.4
- Lec #: 22 · Readings: Section 6.5 and handout
- Lec #: 23 · Readings: Section 6.6 and handout
- Lec #: 24 · Readings: Section 6.7 and handout
- Week #: 9-10 · Lec #: 26 · Readings: Section 8.1 - up to page 374
- Lec #: 27 · Readings: Section 3.4
- Lec #: 28 · Readings: Handout
- Lec #: 29 · Readings: Section 8.2
- Lec #: 30 · Readings: Section 8.3
- Week #: 11-12 · Lec #: 31 · Readings: Sections 7.1, 7.2 and handout
- Lec #: 32 · Readings: Section 7.3 and handout
- Lec #: 33 · Readings: Section 7.6
- Lec #: 34 · Readings: Section 7.7 and handout
- First exam will include material up to approximately this point.
- Summary, chapter 5: Read it, please.
- Second exam will include material up to approximately this point.
- Third exam will include material up to this point (Course end).
- Saff, Edward and Arthur David Snider. Fundamentals of Complex Analysis . 3nd ed. Upper Saddle River: Prentice Hall, 1993. ISBN: 0133274616.
Prerequisites & course information
Prereq: Calculus II (GIR) and ( 18.03 or 18.032 ) U (Spring) 4-0-8 units Credit cannot also be received for 18.075 , 18.0751
Sources
- Course catalogue ↗2026–2027
- MIT OpenCourseWare ↗Spring 2018
- MIT OCW syllabus ↗
- MIT OCW calendar ↗
- MIT OpenCourseWare ↗Fall 1999
- MIT OCW calendar ↗
- MIT OCW readings ↗
- MIT OCW syllabus ↗
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.
- Historical MIT OpenCourseWare offering. Named reading entries and schedule topics are automatically extracted from the linked source pages.