Orthogonal Functions

  • James B. Seaborn


Differential equations that arise in seeking mathematical descriptions of physical systems frequently take the form
$$\Omega u(x)\, = \,\lambda u(x)$$
as we saw in Chapter 7. Here the parameter λ is constant and the operator Ω involves differentiation with respect to x. For different values of λ, the equation has different solutions u(x) We also saw in Chapter 7 that physical constraints may restrict λ to only certain values. We denote these “allowed” values by an index n. To each λ n there corresponds a solution to u n (x) this equation. Typically, these functions obey the relation
$$\int\limits_a^b {u_m ^ * (x)u_n (x)\,dx\, = \,0} \,\,\,\,\text{for}\,m \ne \,n.$$
Functions with this property are called orthogonal functions. They are very important and have wide application in applied mathematics. To illustrate their usefulness, we look at some specific applications, first in quantum mechanics and laer in classical physics. In Section 8.6, we consider a more general treatment of orthogonal functions.


Fourier Series Harmonic Oscillator Classical Physic Arbitrary State Fourier Amplitude 
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Copyright information

© Springer-Verlag New York, Inc. 2002

Authors and Affiliations

  • James B. Seaborn
    • 1
  1. 1.Department of PhysicsUniversity of RichmondUSA

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