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A group of individuals or instances sharing common traits; a category or sort: different kinds of furniture; a new kind of politics. A doubtful or borderline member of a given category: fashioned a kind of shelter; a kind of bluish color.

Archaic a. Underlying character as a determinant of the class to which a thing belongs; nature or essence. Obsolete a. Lineal ancestry or descent. With produce or commodities rather than with money: pay in kind. In the same manner or with an equivalent: returned the slight in kind.

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Usage Note: The words kind, sort, and type can be troublesome when they are used with plural nouns and modifiers. Sentences like I hate these kind of movies may occur with some frequency but are awkward, and some would say, grammatically incorrect.


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The Usage Panel frowns upon these usages. In our survey, 81 percent rejected the use of kind with a plural modifier and plural noun in the sentence Those kind of buildings seem old-fashioned. Fully 88 percent of the Panel found unacceptable the use of kind with a singular modifier and plural noun and verb in That kind of buildings seem old fashioned. In these examples kind would presumably function as a determiner like number in A great number of people have crowded into the lobby. Note that number here is singular, but the plural verb have agrees with the plural noun people, so number is not really the subject of the sentence.

This problem can be avoided by making the phrase entirely singular as in That kind of movie is always enjoyable or by revising so that the noun is the plural subject as in Movies of that kind are always enjoyable. Bear in mind that plural kinds often implies that the phrase refers to a number of different categories of things—more than one genre of movie, for example.

Perhaps the best solution is to drop the kind phrase entirely Those movies are always enjoyable or to be specific Those spy movies are always enjoyable. Kind of. Usage: The mixture of plural and singular constructions, although often used informally with kind and sort , should be avoided in serious writing: children enjoy those kinds not those kind of stories; these sorts not these sort of distinctions are becoming blurred.

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Historically, kind is an unchanged or unmarked plural noun like deer, folk, sheep, and swine, and the construction these kind of is an old one, occurring in the writings of Shakespeare, Swift, Jane Austen, and, in modern times, Winston Churchill and Jimmy Carter. In edited, more formal prose, this kind of and these kinds of are more common. This construction too is often considered incorrect and appears mainly in less formal speech and writing. Switch to new thesaurus. Characterized by kindness and concern for others: altruistic , beneficent , benevolent , benign , benignant , good , goodhearted , kindhearted , kindly.

A class that is defined by the common attribute or attributes possessed by all its members: breed , cast , description , feather , ilk , lot , manner , mold , nature , order , sort , species , stamp , stripe , type , variety. What kind of car is it? He's such a kind man; It was very kind of you to look after the children yesterday. She kindly lent me a handkerchief. Would you kindly stop talking!

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I'll never forget her kindness; Thank you for all your kindness. She is too kind-hearted to hurt an animal. Mentioned in? References in classic literature? Note that the Bessel differential equation is second-order , so there must be two linearly independent solutions. We have found both only for.

For a general nonintegral order, the independent solutions are and. When is an integer , the general real solution is of the form. Neumann function or Weber function , and and are constants. Complex solutions are given by the Hankel functions a. Bessel functions of the third kind. The Bessel functions are orthogonal in according to. Except when is a negative integer ,. In terms of a confluent hypergeometric function of the first kind , the Bessel function is written.

A derivative identity for expressing higher order Bessel functions in terms of is. Asymptotic forms for the Bessel functions are. Abramowitz and Stegun , p.


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  4. The Bessel functions are normalized so that. Integrals involving include. Ratios of Bessel functions of the first kind have continued fraction. The special case of gives as the series. Abramowitz, M.

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    New York: Dover, pp. Arfken, G. Orlando, FL: Academic Press, pp. Cajori, F. A History of Mathematical Notations, Vols. New York: Dover, Hansen, P.

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    Gotha, Lehmer, D. Le Lionnais, F. Les nombres remarquables. Paris: Hermann, Morse, P. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. Spanier, J. Washington, DC: Hemisphere, pp. Wall, H. Analytic Theory of Continued Fractions. New York: Chelsea,