A term (opposed to synthetic; similar to a priori) used by analytic philosophers to describe a proposition that is tautologous due to its form.

Logic: Necessarily true by virtue of the meaning of its component terms alone, without reference to external fact, and with its denial resulting in self-contradiction.

Synonym: tautologous
Compare: synthetic

A complex function f is analytic at z0 if the following conditions are fulfilled:

  1. f is derivable at z0;
  2. There exists a neighborhood V of z0 such that f is derivable at every point of V.

It follows that a complex function is analytic, if these conditions are satisfied for all z0.

analytic language

A language in which inflection plays a minor role -- although no language is purely analytic. The key feature of an analytic language is that it relies on word order to clarify the relationships between words, rather than word-modifying morphemes. The opposite is a synthetic language.

Among the primary synthetic constructions in English is the possessive modifier that allows us to say "The dog's dinner" rather than "The dinner of the dog." Indeed, French lacks for such a modifier, and so this must be expressed as "Le dîner du chien." I'm far from fluent, mind, so please correct me.

Examples of largely analytic languages are Chinese, English and French.

Analytic is a term that describes a complex function that is "differentiable" at and in the neighborhood (region of non-zero area) surrounding some point. A function that is analytic over the entire complex plane is called entire.

But this is an empty definition unless we clarify what it means for a complex function to be differentiable. Differentiability of a complex function f(z = x+iy) is a very strict criterion. f(z) is defined to be differentiable at a point z0 iff the limit as Δz goes to 0 of (f(z0+Δz) - f(z0))/Δz exists and is independent of how Δz goes to 0. For example, the limit must be the same for both the case when Δz = Δx and the case when Δz = iΔy. More generally the limit must be the same for all 360 degrees worth of directions of Δz in the complex plane.

The mathematical necessary conditions for a function f(z) to be differentiable at a point are called the Cauchy-Riemann equations, and are very easy to prove. It can be proven that given any analytic function, its first derivative is analytic as well, implying that every analytic function has derivatives of all orders in the region in which it's analytic. This statement is not true for differentiable functions of real variables. For instance f(x) = x|x| is differentiable at x = 0, but its derivative is not.

Analytic functions have wonderful mathematical properties, and I am unaware of interest in or work on nonanalytic function theory.

Alright, I'm only an amateur linguist, but seeing as it's a fairly basic principle, here goes nothing.

An analytical language is one in which word order has a tremendous effect on meaning, as opposed to a synthetic language, which uses conjugations and other affixes to change meaning and denote the syntactic structure of a sentence.

To compare an analytic language to a synthetic, consider the following sentences in English, an analytic, and Latin, a synthetic.

English
The dog bites the man.

Latin
Canis mordet hominem.

However, if we were to change the word order:

Hominem mordet canis.

The man bites the dog.

The meaning of the sentence in Latin is very nearly the same, if a tad awkward, while in English we are left with a completely different meaning.

This is because Latin uses suffixes to denote parts of a sentence. Canis is the nominative case of the noun canis; while hominem is the accusative case of the noun homo. In other words, canis is doing the acting in the sentence, while homo is being acted upon, requiring a separate conjugation hominem.

In English, however, sentence structure and meaning depends on word placement, as demonstrated above. English lacks separate noun conjugations for each case and instead uses a subject-verb-object(SVO) structure.

Furthermore, topic and emphasis can be changed by changing word order. For example, "I met Mary at the park" can be changed to "In the park, I met Mary." While having the same basic meaning, the corresponding semantic change shifts emphasis from the object, "Mary," to location, "the park."

Other examples of analytical languages include Chinese, which is almost completely analytical, and Thai and Vietnamese.

Sources: Wikipedia, Webster, Latin Made Simple for some quick examples, and my Italian teacher.

Thanks go to wordnerd and Albert Herring for pointing out my woeful linguistic inadequacies.

An`a*lyt"ic (#), An`a*lyt"ic*al (#), a. [Gr. : cf. F. analytique. See Analysis.]

Of or pertaining to analysis; resolving into elements or constituent parts; as, an analytical experiment; analytic reasoning; -- opposed to synthetic.

Analytical or coordinate geometry. See under Geometry. -- Analytic language, a noninflectional language or one not characterized by grammatical endings. -- Analytical table Nat. Hist., a table in which the characteristics of the species or other groups are arranged so as to facilitate the determination of their names.

 

© Webster 1913.

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