The answer by Greg R correctly describes how to register a copyright published under a pseudonym with the US copyright office. It does not make clear that if such a registration is made, anyone who consults the Copyright Office records (which are public) will learn the author's actual name. If that is a problem, one can specify the pseudonym as the "claimant", but then one may have problems verifying authorship and copyright ownership for legal purposes, (as described in the answer by Greg R) should that ever be needed.
In any case, US copyright law 17 USC 401 specifies the proper form of the notice:
(a) General Provisions.—Whenever a work protected under this title is published in the United States or elsewhere by authority of the copyright owner, a notice of copyright as provided by this section may be placed on publicly distributed copies from which the work can be visually perceived, either directly or with the aid of a machine or device.
(b) Form of Notice.—If a notice appears on the copies, it shall consist of the following three elements:
(1) the symbol © (the letter C in a circle), or the word “Copyright”, or the abbreviation “Copr.”; and
(2) the year of first publication of the work; in the case of compilations or derivative works incorporating previously published material, the year date of first publication of the compilation or derivative work is sufficient. The year date may be omitted where a pictorial, graphic, or sculptural work, with accompanying text matter, if any, is reproduced in or on greeting cards, postcards, stationery, jewelry, dolls, toys, or any useful articles; and
(3) the name of the owner of copyright in the work, or an abbreviation by which the name can be recognized, or a generally known alternative designation of the owner.
(c) Position of Notice.—The notice shall be affixed to the copies in such manner and location as to give reasonable notice of the claim of copyright. The Register of Copyrights shall prescribe by regulation, as examples, specific methods of affixation and positions of the notice on various types of works that will satisfy this requirement, but these specifications shall not be considered exhaustive.
(d) Evidentiary Weight of Notice.—If a notice of copyright in the form and position specified by this section appears on the published copy or copies to which a defendant in a copyright infringement suit had access, then no weight shall be given to such a defendant’s interposition of a defense based on innocent infringement in mitigation of actual or statutory damages, except as provided in the last sentence of section 504(2).
So if J Random Author writes a book (or creates some other work) and has it published under the name of J R Doe, valid copyright notices would be:
- Copyright 2022 by J R Doe
- © 2022 by J R Doe
- Copyright © 2022 by J R Doe
- Copyright 2022 J R Doe
- © 2022 J R Doe
- Copyright © 2022 J R Doe
- Copyright 2022 by J Random Author
- © 2022 by J Random Author
- Copyright © 2022 by J Random Author
- Copyright 2022 J Random Author
- © 2022 J Random Author
- Copyright © 2022 J Random Author
Any of these forms will give full legal protection. Most copyright protection is given simply from creating the work and setting it down in a "tangible form" (which includes ink on paper and in a compute file). In the US, registration is required before one can file an infringement suit, and the registration must be before the infringement occurs, or within three months of publication to get statutory damages (often the only useful damages for works not being sold commercially, or not making many sales). The notice itself helps defeat any claim of "innocent infringement (aka 'I didn't know it was protected by copyright") as described in 17 USC 401(d) above. The notice has no other legal effect.