Definition df-lgam 24545

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 24542	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 9813	. . . 4 class ℂ
4		cz 11254	. . . . 5 class ℤ
5		cn 10897	. . . . 5 class ℕ
6	4, 5	cdif 3537	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3537	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1474	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1474	. . . . . . . . . 10 class 𝑚
11		c1 9816	. . . . . . . . . 10 class 1
12		caddc 9818	. . . . . . . . . 10 class +
13	10, 11, 12	co 6549	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 10563	. . . . . . . . 9 class /
15	13, 10, 14	co 6549	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 24105	. . . . . . . 8 class log
17	15, 16	cfv 5804	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 9820	. . . . . . 7 class ·
19	8, 17, 18	co 6549	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 6549	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 6549	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 5804	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 10145	. . . . . 6 class −
24	19, 22, 23	co 6549	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 14264	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 5804	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 6549	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 4643	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1475	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  24562  lgamf  24568  iprodgam  30881