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From Wikipedia

Regular singular point

In mathematics, in the theory of ordinary differential equations in the complex plane \mathbb{C}, the points of \mathbb{C} are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a singularity. Then amongst singular points, an important distinction is made between a regular singular point, where the growth of solutions is bounded (in any small sector) by an algebraic function, and an irregular singular point, where the full solution set requires functions with higher growth rates. This distinction occurs, for example, between the hypergeometric equation, with three regular singular points, and the Bessel equation which is in a sense a limiting case, but where the analytic properties are substantially different.

Formal definitions

More precisely, consider an ordinary linear differential equation of n-th order

\sum_{i=0}^n p_i(z) f^{(i)} (z) = 0

with pi (z) meromorphic functions. One can assume that

p_n(z) = 1. \,

If this is not the case the equation above has to be divided by pn(x). This may introduce singular points to consider.

The equation should be studied on the Riemann sphere to include the point at infinity as a possible singular point. A Möbius transformation may be applied to move ∞ into the finite part of the complex plane if required, see example on Bessel differential equation below.

Then the Frobenius method based on the indicial equation may be applied to find possible solutions that are power series times complex powers (z− a)r near any given a in the complex plane where r need not be an integer; this function may exist, therefore, only thanks to a branch cut extending out from a, or on a Riemann surface of some punctured disc around a. This presents no difficulty for a an ordinary point (Lazarus Fuchs 1866). When a is a regular singular point, which by definition means that

p_{n-i}(z)\,

has a pole of order at most i at a, the Frobenius method also can be made to work and provide n independent solutions near a.

Otherwise the point a is an irregular singularity. In that case the monodromy group relating solutions by analytic continuation has less to say in general, and the solutions are harder to study, except in terms of their asymptotic expansions.

The regularity condition is a kind of Newton polygon condition, in the sense that the allowed poles are in a region, when plotted against i, bounded by a line at 45° to the axes.

An ordinary differential equation whose only singular points, including the point at infinity, are regular singular points is called a Fuchsian ordinary differential equation.

Examples for second order differential equations

In this case the equation above is reduced to:

f(x) + p_1(x) f'(x) + p_0(x) f(x) = 0.\,

One distinguishes the following cases:

  • Point a is an ordinary point when functions p1(x) and p0(x) are analytic at x = a.
  • Point a is a regular singular point if p1(x) has a pole of order 1 at x = a and p0 has a pole of order up to 2 at x = a.
  • Otherwise point a is an irregular singular point.

Listed below are several examples from ordinary differential equations from mathematical physics that have singular points and known solutions.

Bessel differential equation

This is an ordinary differential equation of second order. It is found in the solution to Laplace's equation in cylindrical coordinates:

x^2 \frac{d^2 f}{dx^2} + x \frac{df}{dx} + (x^2 - \alpha^2)f = 0

for an arbitrary real or complex number α (the order of the Bessel function). The most common and important special case is where α is an integern.

Dividing this equation by x2 gives:

\frac{d^2 f}{dx^2} + \frac{1} {x} \frac{df}{dx} + \left (1 - \frac {\alpha^2} {x^2} \right )f = 0

In this case p1(x) = 1/x has a pole of first order at x = 0. When α ≠ 0 p0(x) = (1 − α2/x2) has a pole of second order at x = 0. Thus this equation has a regular singularity at 0.

To see what happens when x→ ∞ one has to use a Möbius transformation, for example x = 1 / (w - b). After performing the algebra:

\frac {d^2 f} {d w^2} + \frac {1} {w-b} \frac {df} {dw} +

\left[ \frac {1} {(w-b)^4} - \frac {\alpha ^2} {(w-b)^2} \right ] f= 0

Now

p1(w) = 1/(w− b)

has a pole of first order at w = b. And p0(w) has a pole of fourth order at w = b. Thus this equation has an irregular singularity w = b corresponding to x at ∞. There is a basis for solutions of this differential equation that are Bessel functions.

Legendre differential equation

This is an ordinary differential equation of second order. It is found in the solution of Laplace's equation in regular verbs, irregular verbs are those verbs that fall outside the standard patterns of conjugation in the languages in which they occur. The idea of an irregular verb is important in second language acquisition, where the verb paradigms of a foreign language are learned systematically, and exceptions listed and carefully noted. Thus for example a school French textbook may have a section at the back listing the French irregular verbs in tables. Irregular verbs are often the most commonly used verbs in the language.

In linguistic analysis, the concept of an irregular verb is most likely to be used in psycholinguistics, and in first-language acquisition studies, where the aim is to establish how the human brain processes its native language. One debate among 20th-century linguists revolved around the question of whether small children learn all verb forms as separate pieces of vocabulary or whether they deduce forms by the application of rules. Since a child can hear a verb for the first time and immediately reuse it correctly in a different tense which he or she has never heard, it is clear that the brain does work with rules, but irregular verbs must be processed differently.

Historical linguists rarely use the category irregular verb. Since most irregularities can be explained historically, these verbs are only irregular when viewed synchronically, not when seen in their historical context.

When languages are being compared informally, one of the few quantitative statistics which are sometimes cited is the number of irregular verbs. These counts are not particularly accurate for a wide variety of reasons, and academic linguists are reluctant to cite them. But it does seem that some languages have a greater tolerance for paradigm irregularity than others.

Prefixed verbs

In English, to withhold conjugates exactly like to hold, and in Spanish, detener ("to detain") conjugates exactly like tener ("to have"). In each case, it is questionable if the compound verb and the main verb are both irregular verbs, or as a single irregular verb, with an optional prefix. The question is compounded by the fact that it is not always predictable if the compound conjugates the same as the base. In Spanish, bendecir ("to bless") conjugates almost exactly like decir ("to say"), but there are significant differences in a few tenses that are impossible to foresee.

Irregular in spelling only

For purposes of psycholinguistics and first language acquisistion studies, only irregularities in the spoken form are relevant. However in the foreign language classroom, the focus can be on the written form, and here irregularities of spelling are equally important.

Some verbs are irregular only in their spelling, but not in their pronunciation. For example, in Spanish, the verb rezar ("to pray") is conjugated in the present subjunctive as rece, reces, rece, etc. The substitution of "c" for "z" does not affect the pronunciation. It is strictly a matter of orthography and can be perfectly predicted (if one knew the rules of Spanish pronunciation and orthography but had never seen the verb "rezar" before, one would still know that the verb would have to be spelled with a "c" in the present subjunctive). Therefore, this verb is not considered to be irregular. Another example of a verb similar to rezar is pagar - to pay. In this verb, g always changes to a gu before an e.

English has similar cases; the verb "pay" sounds regular: "I pay", "I paid", and "I have paid" are all pronounced as expected. But the spelling is irregular and that cannot be perfectly predicted—for example, "pay" and "lay" turn into "paid" and "laid", but "sway" and "stay" turn into "swayed" and "stayed".



From Yahoo Answers

Question:For example, the Spanish verb descansar is a regular verb and the Spanish verb cerrar is an irregular verb. Just by looking at the two verbs alone, I would'nt know which one was regular or irregular when conjugating.

Answers:You can't tell, you have to learn it by heart! However, often verbs which are similar have the same irregularity! E.g cerrar and serrar, both have the e --> ie pattern (Yo cierro and yo sierro) You'll just have to memorize it, just like in English! Or how would you know that to go has an irregular past tense and to walk doesn't ?

Question:My 10year old has homework on shapes, and is asking whats the difference between equalateral and regular shapes. Help!!! its been decades since I even thought about this. I thought both of them had sides of the equal length, so whats the difference?

Answers:Equilateral shapes have all sides of the same length, but a regular polygon has all sides the same length AND all angles the same measure. Most equilateral shapes are also regular shapes. However a rhombus (diamond) for example is equilateral but isn't regular, because not all of the angles are the same.

Question:i need a link to a site that shows two regular polygons with the same area

Answers:How about 3 such polygons ? (To get exactly equal areas, you'd need to use fractions for the sides.)

Question:I am trying to figure out the angle sum on a polygon from my textbook and it says angle sum of a convex polygon, what i want to know is what its the difference between a convex polygon and a normal polygon.

Answers:A convex polygon is what you, and probably everybody else, thinks of as a normal polygon. Polygons with at least four sides can cave inwards and have an angle which is greater than 180 degrees. Draw a five sided polygon with internal angles 100, 80, 70, 240, 60 and you will see what I mean. Polygons with a very large number of sides can have two or more angles of over 180 and look quite weird.

From Youtube

English Tutor - Grammar - Irregular Verbs :This English introductory video shows students the difference between regular and irregular verbs. www.tutorgiant.com

Is a Point Inside or Outside a Regular Polygon? :demonstrations.wolfram.com The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration determines whether a point is inside or outside a polygon by summing the angles between adjacent vertices and the point in question. If the sum is (approximately) 2?, the point is inside; otherwise, it is outside. Contributed by: Jon Perry