Definition df-lgam 26956

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 26953	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11004	. . . 4 class ℂ
4		cz 12468	. . . . 5 class ℤ
5		cn 12125	. . . . 5 class ℕ
6	4, 5	cdif 3894	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3894	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1540	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1540	. . . . . . . . . 10 class 𝑚
11		c1 11007	. . . . . . . . . 10 class 1
12		caddc 11009	. . . . . . . . . 10 class +
13	10, 11, 12	co 7346	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11774	. . . . . . . . 9 class /
15	13, 10, 14	co 7346	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26490	. . . . . . . 8 class log
17	15, 16	cfv 6481	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11011	. . . . . . 7 class ·
19	8, 17, 18	co 7346	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7346	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7346	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6481	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11344	. . . . . 6 class −
24	19, 22, 23	co 7346	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15593	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6481	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7346	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5170	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1541	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  26973  lgamf  26979  iprodgam  35786