Definition df-lgam 26997

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 26994	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11036	. . . 4 class ℂ
4		cz 12500	. . . . 5 class ℤ
5		cn 12157	. . . . 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 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 7368	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11806	. . . . . . . . 9 class /
15	13, 10, 14	co 7368	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26531	. . . . . . . 8 class log
17	15, 16	cfv 6500	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11043	. . . . . . 7 class ·
19	8, 17, 18	co 7368	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7368	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7368	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6500	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11376	. . . . . 6 class −
24	19, 22, 23	co 7368	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15621	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6500	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7368	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5181	. 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  27014  lgamf  27020  iprodgam  35958