Definition df-lgam 26999

Description: Define the log-Gamma function. We can work with this form of the gamma function a bit easier than the equivalent expression for the gamma function itself, and moreover this function is not actually equal to log(Γ(𝑥)) because the branch cuts are placed differently (we do have exp(log Γ(𝑥)) = Γ(𝑥), though). This definition is attributed to Euler, and unlike the usual integral definition is defined on the entire complex plane except the nonpositive integers ℤ ∖ ℕ, where the function has simple poles. (Contributed by Mario Carneiro, 12-Jul-2014.)

Assertion

Ref	Expression
df-lgam	⊢ log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))

Distinct variable group:   𝑧,𝑚

Detailed syntax breakdown of Definition df-lgam

Step	Hyp	Ref	Expression
1		clgam 26996	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11030	. . . 4 class ℂ
4		cz 12518	. . . . 5 class ℤ
5		cn 12168	. . . . 5 class ℕ
6	4, 5	cdif 3887	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3887	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1541	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1541	. . . . . . . . . 10 class 𝑚
11		c1 11033	. . . . . . . . . 10 class 1
12		caddc 11035	. . . . . . . . . 10 class +
13	10, 11, 12	co 7361	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11801	. . . . . . . . 9 class /
15	13, 10, 14	co 7361	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26534	. . . . . . . 8 class log
17	15, 16	cfv 6493	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11037	. . . . . . 7 class ·
19	8, 17, 18	co 7361	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7361	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7361	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6493	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11371	. . . . . 6 class −
24	19, 22, 23	co 7361	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15642	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6493	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7361	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5167	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1542	1 wff log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))

This definition is referenced by:  lgamgulm2  27016  lgamf  27022  iprodgam  35943