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| == '''[[Associated Legendre function]]''' ==
| | {{:{{FeaturedArticleTitle}}}} |
| ----
| | <small> |
| | | ==Footnotes== |
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| In [[mathematics]] and [[physics]], an '''associated Legendre function''' ''P''<sub>''ℓ''</sub><sup>''m''</sup> is related to a [[Legendre polynomial]] ''P''<sub>''ℓ''</sub> by the following equation
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| :<math>
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| P^{m}_\ell(x) = (1-x^2)^{m/2} \frac{d^m P_\ell(x)}{dx^m}, \qquad 0 \le m \le \ell.
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| </math>
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| Although extensions are possible, in this article ''ℓ'' and ''m'' are restricted to integer numbers. For even ''m'' the associated Legendre function is a polynomial, for odd ''m'' the function contains the factor (1−''x'' ² )<sup>½</sup> and hence is not a polynomial.
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| The associated Legendre functions are important in [[quantum mechanics]] and [[potential theory]].
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| According to Ferrers<ref> N. M. Ferrers, ''An Elementary Treatise on Spherical Harmonics'', MacMillan, 1877 (London), p. 77. [http://www.archive.org/stream/elementarytreati00ferriala#page/2/mode/2up Online].</ref> the polynomials were named "Associated Legendre functions" by the British mathematician [[Isaac Todhunter]] in 1875,<ref>I. Todhunter, ''An Elementary Treatise on Laplace's, Lamé's, and Bessel's Functions'', MacMillan, 1875 (London). In fact, Todhunter called the Legendre polynomials "Legendre coefficients". </ref> where "associated function" is Todhunter's translation of the German term ''zugeordnete Function'', coined in 1861 by [[Heinrich Eduard Heine|Heine]],<ref> E. Heine, ''Handbuch der Kugelfunctionen'', G. Reimer, 1861 (Berlin).[http://books.google.com/books?id=YE8DAAAAQAAJ&pg=PA3&dq=Eduard+Heine&hl=en#PPR1,M1 Google book online]</ref> and "Legendre" is in honor of the French mathematician [[Adrien-Marie Legendre]] (1752–1833), who was the first to introduce and study the functions.
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| ===Differential equation===
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| Define
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| :<math>
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| \Pi^{m}_\ell(x) \equiv \frac{d^m P_\ell(x)}{dx^m},
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| </math>
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| where ''P''<sub>''ℓ''</sub>(''x'') is a Legendre polynomial.
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| Differentiating the [[Legendre polynomials#differential equation|Legendre differential equation]]:
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| :<math>
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| (1-x^2) \frac{d^2 \Pi^{0}_\ell(x)}{dx^2} - 2 x \frac{d\Pi^{0}_\ell(x)}{dx} + \ell(\ell+1)
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| \Pi^{0}_\ell(x) = 0,
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| </math>
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| ''m'' times gives an equation for Π<sup>''m''</sup><sub>''l''</sub>
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| :<math>
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| (1-x^2) \frac{d^2 \Pi^{m}_\ell(x)}{dx^2} - 2(m+1) x \frac{d\Pi^{m}_\ell(x)}{dx} + \left[\ell(\ell+1)
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| -m(m+1) \right] \Pi^{m}_\ell(x) = 0 .
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| </math>
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| After substitution of
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| :<math>
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| \Pi^{m}_\ell(x) = (1-x^2)^{-m/2} P^{m}_\ell(x),
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| </math>
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| and after multiplying through with <math>(1-x^2)^{m/2}</math>, we find the ''associated Legendre differential equation'':
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| :<math>
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| (1-x^2) \frac{d^2 P^{m}_\ell(x)}{dx^2} -2x\frac{d P^{m}_\ell(x)}{dx} +
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| \left[ \ell(\ell+1) - \frac{m^2}{1-x^2}\right] P^{m}_\ell(x)= 0 .
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| </math>
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| One often finds the equation written in the following equivalent way
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| :<math>
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| \left( (1-x^{2})\; y\,' \right)' +\left( \ell(\ell+1)
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| -\frac{m^{2} }{1-x^{2} } \right) y=0,
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| </math>
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| where the primes indicate differentiation with respect to ''x''.
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| In physical applications it is usually the case that ''x'' = cosθ, then the associated Legendre differential equation takes the form
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| :<math>
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| \frac{1}{\sin \theta}\frac{d}{d\theta} \sin\theta \frac{d}{d\theta}P^{m}_\ell
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| +\left[ \ell(\ell+1) - \frac{m^2}{\sin^2\theta}\right] P^{m}_\ell = 0.
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| </math>
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| ''[[Associated Legendre function|.... (read more)]]''
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| {| class="wikitable collapsible collapsed" style="width: 90%; float: center; margin: 0.5em 1em 0.8em 0px;"
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| |-
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| ! style="text-align: center;" | [[Associated Legendre function#Reference|notes]]
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| |-
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| {{reflist|2}} | | {{reflist|2}} |
| |}
| | </small> |
After decades of failure to slow the rising global consumption of coal, oil and gas,[1] many countries have proceeded as of 2024 to reconsider nuclear power in order to lower the demand for fossil fuels.[2] Wind and solar power alone, without large-scale storage for these intermittent sources, are unlikely to meet the world's needs for reliable energy.[3][4][5] See Figures 1 and 2 on the magnitude of the world energy challenge.
Nuclear power plants that use nuclear reactors to create electricity could provide the abundant, zero-carbon, dispatchable[6] energy needed for a low-carbon future, but not by simply building more of what we already have. New innovative designs for nuclear reactors are needed to avoid the problems of the past.
(CC) Image: Geoff Russell Fig.1 Electricity consumption may soon double, mostly from coal-fired power plants in the developing world.
[7] Issues Confronting the Nuclear Industry
New reactor designers have sought to address issues that have prevented the acceptance of nuclear power, including safety, waste management, weapons proliferation, and cost. This article will summarize the questions that have been raised and the criteria that have been established for evaluating these designs. Answers to these questions will be provided by the designers of these reactors in the articles on their designs. Further debate will be provided in the Discussion and the Debate Guide pages of those articles.
- ↑ Global Energy Growth by Our World In Data
- ↑ Countries, organizations, and public figures that have reconsidered their stance on nuclear power are listed on the External Links tab of this article.
- ↑ Pumped storage is currently the most economical way to store electricity, but it requires a large reservoir on a nearby hill or in an abandoned mine. Li-ion battery systems at $500 per KWh are not practical for utility-scale storage. See Energy Storage for a summary of other alternatives.
- ↑ Utilities that include wind and solar power in their grid must have non-intermittent generating capacity (typically fossil fuels) to handle maximum demand for several days. They can save on fuel, but the cost of the plant is the same with or without intermittent sources.
- ↑ Mark Jacobson believes that long-distance transmission lines can provide an alternative to costly storage. See the bibliography for more on this proposal and the critique by Christopher Clack.
- ↑ "Load following" is the term used by utilities, and is important when there is a lot of wind and solar on the grid. Some reactors are not able to do this.
- ↑ Fig.1.3 in Devanney "Why Nuclear Power has been a Flop"
