Definition df-lgam 26981

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 26978	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11127	. . . 4 class ℂ
4		cz 12588	. . . . 5 class ℤ
5		cn 12240	. . . . 5 class ℕ
6	4, 5	cdif 3923	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3923	. . 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 11130	. . . . . . . . . 10 class 1
12		caddc 11132	. . . . . . . . . 10 class +
13	10, 11, 12	co 7405	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11894	. . . . . . . . 9 class /
15	13, 10, 14	co 7405	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26515	. . . . . . . 8 class log
17	15, 16	cfv 6531	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11134	. . . . . . 7 class ·
19	8, 17, 18	co 7405	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7405	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7405	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6531	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11466	. . . . . 6 class −
24	19, 22, 23	co 7405	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15702	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6531	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7405	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5201	. 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  26998  lgamf  27004  iprodgam  35759