An Etymological Dictionary of Astronomy and Astrophysics

1a) To change (something) from a smaller form and/or size to a larger one.

1b) To increase the extent, number, volume or scope of (something).

1c) To express (something) at length and/or in detail.

1d) Math.: To write (a mathematical expression) as a sum of terms in an extended form.

2. to increase or grow in extent, bulk, scope; to spread out; unfold; develop.
   

See also → spread, → extend, → dilate.

Etymology (EN): M.E. expanden, from O.Fr. espandre “to spread, spread out,” from L. expandere “to spread out, unfold, expand,” from → ex- “out” + pandere “to spread, stretch,” from nasalized form of PIE root *pete- “to spread,” a Pers. offshoot of which is pahn “wide, broad,” as below.
from

Etymology (PE): Sopândan, “to stretch, to extend, open out” from so- “out,” → ex-, + pân, from pan, variant of pahn “wide, large, broad, ample, extensive,” from Mid.Pers. pah(a)n; Av. paθana- “broad, wide, spacious;” PIE root *pete- “to spread;” cf. L. patere “to be open,”
Gk. petannynai “to spread out,” petalon “a leaf.”

Alternatively, sopândan “to open up, to open out,” from so- “out,” → ex-, + pân “open,” cf. Yaghnobi pen, peyn “to open, unroll,” Sogd. pyn- “to open,” ultimately from Proto-Ir. *apa- “off, away, from” + *hHai- “to chain, bind,” from which Pers. gošâdan “to open, release” (Cheung 2007).