Definition df-lgam 26984

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 26981	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11038	. . . 4 class ℂ
4		cz 12526	. . . . 5 class ℤ
5		cn 12176	. . . . 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 11041	. . . . . . . . . 10 class 1
12		caddc 11043	. . . . . . . . . 10 class +
13	10, 11, 12	co 7369	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11809	. . . . . . . . 9 class /
15	13, 10, 14	co 7369	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26520	. . . . . . . 8 class log
17	15, 16	cfv 6500	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11045	. . . . . . 7 class ·
19	8, 17, 18	co 7369	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7369	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7369	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6500	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11379	. . . . . 6 class −
24	19, 22, 23	co 7369	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15650	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6500	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7369	. . 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  27001  lgamf  27007  iprodgam  35926