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Differential Operator -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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The operator representing the computation of a derivative,

D^~=d/(dx),

(1)

sometimes also called the Newton-Leibniz operator. The second derivative is then denoted D^~^2, the third D^~^3, etc. The integral is denoted D^~^(-1).

The differential operator satisfies the identity

(2x-d/(dx))^n1=H_n(x),

(2)

where H_n(x) is a Hermite polynomial (Arfken 1985, p. 718), where the first few cases are given explicitly by

The symbol theta can be used to denote the operator

theta=xd/(dx)

(9)

(Bailey 1935, p. 8). A fundamental identity for this operator is given by

(xD^~)^n=sum_(k=0)^nS(n,k)x^kD^~^k,

(10)

where S(n,k) is a Stirling number of the second kind (Roman 1984, p. 144), giving

and so on (OEIS A008277). Special cases include

A shifted version of the identity is given by

[(x-a)D^~]^n=sum_(k=0)^nS(n,k)(x-a)^kD^~^k

(18)

(Roman 1984, p. 146).

See also

Convective Derivative, Derivative, Fractional Derivative, Gradient

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: University Press, 1935.Roman, S. The Umbral Calculus. New York: Academic Press, pp. 59-63, 1984.Sloane, N. J. A. Sequence A008277 in "The On-Line Encyclopedia of Integer Sequences."

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Differential Operator

Cite this as:

Weisstein, Eric W. "Differential Operator." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DifferentialOperator.html

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