Definition df-lgam 26927

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 26924	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11007	. . . 4 class ℂ
4		cz 12471	. . . . 5 class ℤ
5		cn 12128	. . . . 5 class ℕ
6	4, 5	cdif 3900	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3900	. . 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 11010	. . . . . . . . . 10 class 1
12		caddc 11012	. . . . . . . . . 10 class +
13	10, 11, 12	co 7349	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11777	. . . . . . . . 9 class /
15	13, 10, 14	co 7349	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26461	. . . . . . . 8 class log
17	15, 16	cfv 6482	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11014	. . . . . . 7 class ·
19	8, 17, 18	co 7349	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7349	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7349	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6482	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11347	. . . . . 6 class −
24	19, 22, 23	co 7349	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15593	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6482	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7349	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5173	. 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  26944  lgamf  26950  iprodgam  35715