Definition df-lgam 25604

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 25601	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 10524	. . . 4 class ℂ
4		cz 11969	. . . . 5 class ℤ
5		cn 11625	. . . . 5 class ℕ
6	4, 5	cdif 3878	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3878	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1537	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1537	. . . . . . . . . 10 class 𝑚
11		c1 10527	. . . . . . . . . 10 class 1
12		caddc 10529	. . . . . . . . . 10 class +
13	10, 11, 12	co 7135	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11286	. . . . . . . . 9 class /
15	13, 10, 14	co 7135	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25146	. . . . . . . 8 class log
17	15, 16	cfv 6324	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 10531	. . . . . . 7 class ·
19	8, 17, 18	co 7135	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7135	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7135	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6324	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 10859	. . . . . 6 class −
24	19, 22, 23	co 7135	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15034	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6324	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7135	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5110	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1538	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  25621  lgamf  25627  iprodgam  33087