Definition df-lgam 27000

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 26997	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11027	. . . 4 class ℂ
4		cz 12515	. . . . 5 class ℤ
5		cn 12165	. . . . 5 class ℕ
6	4, 5	cdif 3880	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3880	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1546	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1546	. . . . . . . . . 10 class 𝑚
11		c1 11030	. . . . . . . . . 10 class 1
12		caddc 11032	. . . . . . . . . 10 class +
13	10, 11, 12	co 7356	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11798	. . . . . . . . 9 class /
15	13, 10, 14	co 7356	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26536	. . . . . . . 8 class log
17	15, 16	cfv 6485	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11034	. . . . . . 7 class ·
19	8, 17, 18	co 7356	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7356	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7356	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6485	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11368	. . . . . 6 class −
24	19, 22, 23	co 7356	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15639	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6485	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7356	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5153	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1547	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  27017  lgamf  27023  iprodgam  35970