Definition df-lgam 26936

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 26933	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11073	. . . 4 class ℂ
4		cz 12536	. . . . 5 class ℤ
5		cn 12193	. . . . 5 class ℕ
6	4, 5	cdif 3914	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3914	. . 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 11076	. . . . . . . . . 10 class 1
12		caddc 11078	. . . . . . . . . 10 class +
13	10, 11, 12	co 7390	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11842	. . . . . . . . 9 class /
15	13, 10, 14	co 7390	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26470	. . . . . . . 8 class log
17	15, 16	cfv 6514	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11080	. . . . . . 7 class ·
19	8, 17, 18	co 7390	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7390	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7390	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6514	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11412	. . . . . 6 class −
24	19, 22, 23	co 7390	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15659	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6514	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7390	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5191	. 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  26953  lgamf  26959  iprodgam  35736