Definition df-lgam 26523

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 26520	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11108	. . . 4 class ℂ
4		cz 12558	. . . . 5 class ℤ
5		cn 12212	. . . . 5 class ℕ
6	4, 5	cdif 3946	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3946	. . 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 11111	. . . . . . . . . 10 class 1
12		caddc 11113	. . . . . . . . . 10 class +
13	10, 11, 12	co 7409	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11871	. . . . . . . . 9 class /
15	13, 10, 14	co 7409	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26063	. . . . . . . 8 class log
17	15, 16	cfv 6544	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11115	. . . . . . 7 class ·
19	8, 17, 18	co 7409	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7409	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7409	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6544	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11444	. . . . . 6 class −
24	19, 22, 23	co 7409	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15632	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6544	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7409	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5232	. 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  26540  lgamf  26546  iprodgam  34743