Definition df-lgam 26982

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 26979	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11036	. . . 4 class ℂ
4		cz 12524	. . . . 5 class ℤ
5		cn 12174	. . . . 5 class ℕ
6	4, 5	cdif 3886	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3886	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1541	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1541	. . . . . . . . . 10 class 𝑚
11		c1 11039	. . . . . . . . . 10 class 1
12		caddc 11041	. . . . . . . . . 10 class +
13	10, 11, 12	co 7367	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11807	. . . . . . . . 9 class /
15	13, 10, 14	co 7367	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26518	. . . . . . . 8 class log
17	15, 16	cfv 6498	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11043	. . . . . . 7 class ·
19	8, 17, 18	co 7367	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7367	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7367	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6498	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11377	. . . . . 6 class −
24	19, 22, 23	co 7367	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15648	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6498	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7367	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5166	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1542	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  26999  lgamf  27005  iprodgam  35924