“rAghava-simha” is a unique form combining qualities of rAma & narasimha

“rAghava-simha” is a unique form combining qualities of rAma & narasimha. Let me explain the Kaivalyopanishad mantra recommending rAghava-simha dhyAna.

उमासहायं परमेश्वरं प्रभुं त्रिलोचनं नीलकण्ठं प्रशान्तम् । ध्यात्वा मुनिर्गच्छति भूतयोनिं समस्तसाक्षिं तमसः परस्तात्+

उमासहायं परमेश्वरं [One whose helper is lakShmI, whose supreme rulership is made fruitful by her]

“उमा” is a name of lakShmI here and not pArvati, as tApanIya upanishad calls narasimha as “उमापति”. The root comes from अव - to protect - she is a protector (mediator)+

“उमासहायं” - vedAntAchArya calls shrI “rakShaNa-hetu” - she, his helper, induces him to protect.

“परमेश्वरं” does not just mean “supreme ruler” but that his rulership is fruitful due to shrI. Parashara Bhattar says she embraces him so that he as “परमेश्वर”, resolves to protect+

प्रभुं त्रिलोचनं नीलकण्ठं प्रशान्तम् [Who is most powerful even in his avatArAs, who is the eye of the three worlds in his avatArAs, who has a pure, dark blue form as rAma, who is tranquil after establishing dharma]. This set of names shows his accessibility, having been precipitated by shrI devI’s mediation. He becomes manifest in the form of taking avatAra among us, to be visible to us+

“प्रभुं” - Despite taking births, he is still the master of all, ie, he is not diminished like us. narasimha assumed the demeanor of a lion & acted like one, but he had one staunch awareness - “It is my nature to protect prahlAda”+

“त्रिलोचनं” - Does not mean “3-eyed”. It means, “One who is the eye of the 3 worlds or who illumines the 3 worlds” - ie, all can see his form in his avatArAs, signifying his accessibility.

The example of his accessibility is stated by the name “नीलकण्ठं” as below. “नीलकण्ठं” -  “कण्ठं” does not mean throat. It means “that which emits sound”. vyAkaraNa says “अहविसर्जनीयाः कण्ठे” - That which emits sound like अ, ह etc. bhagavAn’s body is composed of the 3 Vedas that constitute “sound” being “emitted” - ie, shines out to the meditators, Thus it is said,

सर्वशक्तिः पुरा विस्नोः र्क् यजुस् सामस्-संज़ित । सैष त्रयी तपत्यंहो जगतश्र हिंसति या  (~vishNu purAna 2.11.7)[The form of vishNu which is “सर्वशक्तिः” is signified by Rik, Yajus & Saman; this trayI burns all sins & that resides resplendent in the Sun.]. Hence, “कण्ठं” refers to his form as the abode of the 3 Vedas. “नीलकण्ठं” means it is a dark blue form -

“नीलमेघश्याम” - a dark blue cloud, which cools the jIvAs scorched by samsAra.

This refers to rAma as vAlmiki rAmAyaNa says “मेघश्यामं महाबाहुं” - a dark form seen by all+

narasimha who is “उमासहायं”, “परमेश्वरं”, “प्रभुं” & “त्रिलोचनं” granted joy to all with a dark form as rAma (नीलकण्ठं). rAghava-simha tattva is being alluded to here. As rAma had a human-like form, Upanishad says his dark form is “कण्ठं” - वेदमय - not made of flesh & blood.

“प्रशान्तम्” - He becomes calm only after taking birth among devotees & displaying his beautiful form to dispel their distress. 

Narasimha was pacified only when he was sure prahlAda was safe; rAma was cured of fever only when vibhIShaNa became King (तदा रामो विज्वरः प्रमुमोद)+

nAradIya purANa has an upabrahmaNa for this mantra: श्रीमन्नृकेसरितनो जगदेकबंन्धो श्रीनीलकण्ठ करुणार्णवे

[Husband of shrI, with the body of half-man, half lion, sole relative of the Universe, with a beautiful, non-material, dark-blue form as rAma and an ocean of mercy]+