Definition df-lgam 24745

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 24742	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 9934	. . . 4 class ℂ
4		cz 11377	. . . . 5 class ℤ
5		cn 11020	. . . . 5 class ℕ
6	4, 5	cdif 3571	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3571	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1482	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1482	. . . . . . . . . 10 class 𝑚
11		c1 9937	. . . . . . . . . 10 class 1
12		caddc 9939	. . . . . . . . . 10 class +
13	10, 11, 12	co 6650	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 10684	. . . . . . . . 9 class /
15	13, 10, 14	co 6650	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 24301	. . . . . . . 8 class log
17	15, 16	cfv 5888	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 9941	. . . . . . 7 class ·
19	8, 17, 18	co 6650	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 6650	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 6650	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 5888	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 10266	. . . . . 6 class −
24	19, 22, 23	co 6650	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 14416	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 5888	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 6650	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 4729	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1483	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  24762  lgamf  24768  iprodgam  31628