Definition df-lgam 25596

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 25593	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 10535	. . . 4 class ℂ
4		cz 11982	. . . . 5 class ℤ
5		cn 11638	. . . . 5 class ℕ
6	4, 5	cdif 3933	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3933	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1536	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1536	. . . . . . . . . 10 class 𝑚
11		c1 10538	. . . . . . . . . 10 class 1
12		caddc 10540	. . . . . . . . . 10 class +
13	10, 11, 12	co 7156	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11297	. . . . . . . . 9 class /
15	13, 10, 14	co 7156	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25138	. . . . . . . 8 class log
17	15, 16	cfv 6355	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 10542	. . . . . . 7 class ·
19	8, 17, 18	co 7156	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7156	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7156	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6355	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 10870	. . . . . 6 class −
24	19, 22, 23	co 7156	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15042	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6355	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7156	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5146	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1537	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  25613  lgamf  25619  iprodgam  32974