2016 Applied Analysis II

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Academic unit or major
Undergraduate major in Mathematics
Yoneda Tsuyoshi  Yoneda Tsuyoshi  Onodera Michiaki 
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Course description and aims

In this course, we re-consider Fourier series in the functional analysis framework, and we explain important inequalities and orthonormal system. We also explain definition and properties of Fourier transform which can be obtained as continuous limit of Fourier series. Lastly we consider several partial differential equations as applications of Fourier series and Fourier transform. This course is a continuation of Applied Analysis I.

The notions of Fourier series and Fourier transform are fundamental not only in mathematics but also in science, and are applicable to describe a wide variety of objects. On the other hand, these abstract notions are not easy to comprehend without suitable training. To that end, rigorous proofs will be given for most propositions, lemmas and theorems. Moreover we study the most fundamental PDEs such as wave, heat and Laplace's equations.

Student learning outcomes

Students are expected to study basic concepts of complex analysis. More precisely, we study Fourier series, Fourier transform, these definitions, properties and method of these calculations. These have important roles in science and technology.


Bessel's inequality, Parseval's equality, Fourier transform, wave equation, heat equation, Laplace's equation

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
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Class flow

Before coming to class, students should read the course schedule and check what topics will be covered.
Required learning should be completed outside of the classroom for preparation and review purposes.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Function spaces, Orthonormal system Details will be provided during each class session
Class 2 Bessel's inequality, Parseval's equality
Class 3 Fourier 's integral formula
Class 4 Fourier transform and Fourier inverse transform
Class 5 properties of Fourier transform
Class 6 application to wave equations
Class 7 application to heat equations
Class 8 application to Laplace's equation, comprehension check-up


None required

Reference books, course materials, etc.

None required

Assessment criteria and methods

Students' course scores are based on final exam 50% and midterm exam 50%.

Related courses

  • ZUA.C201 : Advanced Calculus I
  • ZUA.C203 : Advanced Calculus II
  • MTH.C211 : Applied Analysis I
  • MTH.C301 : Complex Analysis I
  • MTH.C302 : Complex Analysis II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Calculus I/Recitation and Calculus II+ Recitation.

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