The mean of the Tracy–Widom distribution

Find a closed form for the mean μ ≈ −1.7710868074 of the Tracy–Widom distribution of the largest GUE eigenvalue.

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1	{
2	"domain": "physics-astronomy",
3	"topics": [
4	"mathematical-physics",
5	"probability"
6	],
7	"type": "construction",
8	"source": {
9	"key": "horizonmath",
10	"name": "HorizonMath",
11	"url": "https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L660-L670",
12	"id": "tracy_widom_f2_mean",
13	"curator": "Erik Y. Wang, Sumeet R. Motwani et al.",
14	"status": "open",
15	"as_of": "2026-10-20",
16	"catalog": "0521c19bbd38"
17	},
18	"review": true,
19	"approval": "first-time"
20	}