Properties of Differential Operator
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Theorem
The differential operator $D$ has the following properties:
| \(\text {(1)}: \quad\) | \(\ds \dfrac {\map f x} D\) | \(=\) | \(\ds \int \map f x \rd x\) | |||||||||||
| \(\text {(2)}: \quad\) | \(\ds \dfrac {x^n} {\paren {D + p}^q}\) | \(=\) | \(\ds \paren {1 + \dfrac D p}^{-q} \dfrac {x^n} {p^q}\) | |||||||||||
| \(\text {(3)}: \quad\) | \(\ds \map F D e^{a x}\) | \(=\) | \(\ds e^{a x} \map F a\) | |||||||||||
| \(\text {(4)}: \quad\) | \(\ds \map F D e^{a x} \map f x\) | \(=\) | \(\ds e^{a x} \map F {D + a} \map f x\) | |||||||||||
| \(\text {(5)}: \quad\) | \(\ds \map F {D^2} \sin a x\) | \(=\) | \(\ds \map F {-a^2} \sin a x\) | |||||||||||
| \(\text {(6)}: \quad\) | \(\ds \map F {D^2} \cos a x\) | \(=\) | \(\ds \map F {-a^2} \cos a x\) |
Proof
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Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): differential operator
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): differential operator