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Consider two examples:

  1. Gamma function written as Γ(n);
  2. gamma-butyrolactone written as γ-butyrolactone.

Is there a rule which says how the lowercase and capital Greek letters must be reflected when they are written in English? E.g. in the middle of a sentence capital Γ must be written as "Gamma", whereas γ -- as "gamma" only?

If not, how does one know for sure whether it is supposed to be a capital Greek letter or not, when only an English word for it is provided?

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    You need to provide definitions for these terms (so we know what they are) along with some actual usages in context. As written, it can be assumed they follow the accepted convention for writing these technical terms. – user3169 Jul 12 '17 at 21:10
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Questions about the finer points of capitalization are a matter of style; there is no universal consensus on what to capitalize or not aside from very basic conventions. For example, the standard is to capitalize all the important words in the title of a book; there is no agreement, however, as to which words should be considered important or not.

Your editor, publication, or organization will set their own guidelines, and in the absence of direct instruction, you should adhere to the discipline of an appropriate style manual if striving for consistency.

All that said,

  1. The presentation of glyph or other symbol, generally, has no bearing whatsoever on the representation of its name. That these functions are signified using the Greek alphabet, which has uppercase and lowercase letters, is incidental. If they were represented with a special symbol, or a numeral, or a Chinese character, the name of the function would remain the same, and you would capitalize them or not depending on the general rules of your style.

  2. When writing in English, you follow the conventions of English. English is no more mathematical notation than it is German or French, and you would not capitalize a reference to mädchen just because nouns are capitalized in written German, and you would not use guillemets to quote a passage by Proust just because that is how quotations are represented in written French.

The very article you link refers exclusively to gamma functions.

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When letters from foreign alphabets are used as symbols in English -- like Greek π or Hebrew ℵ -- there is a conventional case assigned to that symbol. Like π for 3.1415926... is always written in lower case, even if it's at the beginning of a sentence.

When you are using English words for the foreign letters, like "pi" instead of the actual Greek symbol -- as I am forced to do here because I don't know how to enter foreign letters -- then normal English capitalization rules apply to that word. Like I would type, "The symbol for the ratio of circumference to diameter is pi", but, "Pi is the symbol for ..." etc.

When you are writing actual words and sentences, the normal capitalization rules (for that language) apply.

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