Definition df-lgam 25703

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 25700	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 10573	. . . 4 class ℂ
4		cz 12020	. . . . 5 class ℤ
5		cn 11674	. . . . 5 class ℕ
6	4, 5	cdif 3855	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3855	. . 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 10576	. . . . . . . . . 10 class 1
12		caddc 10578	. . . . . . . . . 10 class +
13	10, 11, 12	co 7150	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11335	. . . . . . . . 9 class /
15	13, 10, 14	co 7150	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25245	. . . . . . . 8 class log
17	15, 16	cfv 6335	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 10580	. . . . . . 7 class ·
19	8, 17, 18	co 7150	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7150	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7150	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6335	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 10908	. . . . . 6 class −
24	19, 22, 23	co 7150	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15090	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6335	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7150	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5112	. 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  25720  lgamf  25726  iprodgam  33223