Lesson plan
Singularity function
Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. The functions are notated with brackets, as \langle x-a\rangle ^n where n is an integer. The " \langle \rangle " are often referred to as singularity brackets. The functions are defined as: : }{dx^ }\delta(x-a)\, |- | -2 | \frac{d}{dx}\delta(x-a)\, |- | -1 | \delta(x-a)\, |- | 0 | H(x-a)\, |- | 1 | (x-a)H(x-a)\, |- | 2 | (x-a)^2H(x-a) |- | \ge 0 | (x-a)^nH(x-a) |} where: is the Dirac delta function, also called the unit impulse. The first derivative of is also called the unit doublet. The function H(x) is the Heaviside step function: for and for . The value of will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for since the functions contain a multiplicative factor of for . \langle x-a\rangle^1 is also called the ramp function.
Article summary
Singularity functions are a class of discontinuous functions that contain singularities, i.e., they are discontinuous at their singular points. Singularity functions have been heavily studied in the field of mathematics under the alternative names of generalized functions and distribution theory. The functions are notated with brackets, as \langle x-a\rangle ^n where n is an integer. The " \langle \rangle " are often referred to as singularity brackets. The functions are defined as: : }{dx^ }\delta(x-a)\, |- | -2 | \frac{d}{dx}\delta(x-a)\, |- | -1 | \delta(x-a)\, |- | 0 | H(x-a)\, |- | 1 | (x-a)H(x-a)\, |- | 2 | (x-a)^2H(x-a) |- | \ge 0 | (x-a)^nH(x-a) |} where: is the Dirac delta function, also called the unit impulse. The first derivative of is also called the unit doublet. The function H(x) is the Heaviside step function: for and for . The value of will depend upon the particular convention chosen for the Heaviside step function. Note that this will only be an issue for since the functions contain a multiplicative factor of for . \langle x-a\rangle^1 is also called the ramp function.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.