Definition df-lgam 26929

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 26926	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11066	. . . 4 class ℂ
4		cz 12529	. . . . 5 class ℤ
5		cn 12186	. . . . 5 class ℕ
6	4, 5	cdif 3911	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3911	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1539	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1539	. . . . . . . . . 10 class 𝑚
11		c1 11069	. . . . . . . . . 10 class 1
12		caddc 11071	. . . . . . . . . 10 class +
13	10, 11, 12	co 7387	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11835	. . . . . . . . 9 class /
15	13, 10, 14	co 7387	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26463	. . . . . . . 8 class log
17	15, 16	cfv 6511	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11073	. . . . . . 7 class ·
19	8, 17, 18	co 7387	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7387	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7387	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6511	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11405	. . . . . 6 class −
24	19, 22, 23	co 7387	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15652	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6511	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7387	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5188	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1540	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  26946  lgamf  26952  iprodgam  35729