Definition df-lgam 26073

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 26070	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 10800	. . . 4 class ℂ
4		cz 12249	. . . . 5 class ℤ
5		cn 11903	. . . . 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 1538	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1538	. . . . . . . . . 10 class 𝑚
11		c1 10803	. . . . . . . . . 10 class 1
12		caddc 10805	. . . . . . . . . 10 class +
13	10, 11, 12	co 7255	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11562	. . . . . . . . 9 class /
15	13, 10, 14	co 7255	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25615	. . . . . . . 8 class log
17	15, 16	cfv 6418	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 10807	. . . . . . 7 class ·
19	8, 17, 18	co 7255	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7255	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7255	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6418	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11135	. . . . . 6 class −
24	19, 22, 23	co 7255	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15325	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6418	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7255	. . 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 1539	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  26090  lgamf  26096  iprodgam  33614