Jacobi Differential Equation
المؤلف:
Bliss, G. A
المصدر:
alculus of Variations. Chicago, IL: Open Cour
الجزء والصفحة:
...
13-6-2018
1843
Jacobi Differential Equation
 |
(1)
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or
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(2)
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The solutions are Jacobi polynomials 
or, in terms of hypergeometric functions, as
 |
(3)
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The equation (2) can be transformed to
![(d^2u)/(dx^2)+[1/4(1-alpha^2)/((1-x)^2)+1/4(1-beta^2)/((1+x)^2)+(n(n+alpha+beta+1)+1/2(alpha+1)(beta+1))/(1-x^2)]u=0,](http://mathworld.wolfram.com/images/equations/JacobiDifferentialEquation/NumberedEquation4.gif) |
(4)
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where
 |
(5)
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and
![(d^2u)/(dtheta^2)+[(1/4-alpha^2)/(4sin^2(1/2theta))+(1/4-beta^2)/(4cos^2(1/2theta))+(n+(alpha+beta+1)/2)^2]u=0,](http://mathworld.wolfram.com/images/equations/JacobiDifferentialEquation/NumberedEquation6.gif) |
(6)
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where
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(7)
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Zwillinger (1997, p. 123) gives a related differential equation he terms Jacobi's equation
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(8)
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(Iyanaga and Kawada 1980, p. 1480), which has solution
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(9)
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Zwillinger (1997, p. 120; duplicated twice) also gives another types of ordinary differential equation called a Jacobi equation,
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(10)
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(Ince 1956, p. 22).
In the calculus of variations, the partial differential equation
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(11)
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where
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(12)
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is called the Jacobi differential equation.
REFERENCES:
Bliss, G. A. Calculus of Variations. Chicago, IL: Open Court, pp. 162-163, 1925.
Ince, E. L. Ordinary Differential Equations. New York: Dover, p. 22, 1956.
Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 1480, 1980.
Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 120, 1997.
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