Definition df-lgam 26168

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 26165	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 10869	. . . 4 class ℂ
4		cz 12319	. . . . 5 class ℤ
5		cn 11973	. . . . 5 class ℕ
6	4, 5	cdif 3884	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3884	. . 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 10872	. . . . . . . . . 10 class 1
12		caddc 10874	. . . . . . . . . 10 class +
13	10, 11, 12	co 7275	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11632	. . . . . . . . 9 class /
15	13, 10, 14	co 7275	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25710	. . . . . . . 8 class log
17	15, 16	cfv 6433	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 10876	. . . . . . 7 class ·
19	8, 17, 18	co 7275	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7275	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7275	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6433	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11205	. . . . . 6 class −
24	19, 22, 23	co 7275	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15397	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6433	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7275	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5157	. 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  26185  lgamf  26191  iprodgam  33708