Definition df-lgam 27062

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 27059	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11153	. . . 4 class ℂ
4		cz 12613	. . . . 5 class ℤ
5		cn 12266	. . . . 5 class ℕ
6	4, 5	cdif 3948	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3948	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1539	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1539	. . . . . . . . . 10 class 𝑚
11		c1 11156	. . . . . . . . . 10 class 1
12		caddc 11158	. . . . . . . . . 10 class +
13	10, 11, 12	co 7431	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11920	. . . . . . . . 9 class /
15	13, 10, 14	co 7431	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26596	. . . . . . . 8 class log
17	15, 16	cfv 6561	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11160	. . . . . . 7 class ·
19	8, 17, 18	co 7431	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7431	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7431	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6561	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11492	. . . . . 6 class −
24	19, 22, 23	co 7431	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15722	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6561	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7431	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5225	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1540	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  27079  lgamf  27085  iprodgam  35742