Definition df-lgam 27077

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 27074	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11151	. . . 4 class ℂ
4		cz 12611	. . . . 5 class ℤ
5		cn 12264	. . . . 5 class ℕ
6	4, 5	cdif 3960	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3960	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1536	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1536	. . . . . . . . . 10 class 𝑚
11		c1 11154	. . . . . . . . . 10 class 1
12		caddc 11156	. . . . . . . . . 10 class +
13	10, 11, 12	co 7431	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11918	. . . . . . . . 9 class /
15	13, 10, 14	co 7431	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26611	. . . . . . . 8 class log
17	15, 16	cfv 6563	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11158	. . . . . . 7 class ·
19	8, 17, 18	co 7431	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7431	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7431	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6563	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11490	. . . . . 6 class −
24	19, 22, 23	co 7431	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15719	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6563	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7431	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5231	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1537	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  27094  lgamf  27100  iprodgam  35722