Definition df-lgam 27358

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 27355	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11208	. . . 4 class ℂ
4		cz 12703	. . . . 5 class ℤ
5		cn 12345	. . . . 5 class ℕ
6	4, 5	cdif 4047	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 4047	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1568	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1568	. . . . . . . . . 10 class 𝑚
11		c1 11211	. . . . . . . . . 10 class 1
12		caddc 11213	. . . . . . . . . 10 class +
13	10, 11, 12	co 7427	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11983	. . . . . . . . 9 class /
15	13, 10, 14	co 7427	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26894	. . . . . . . 8 class log
17	15, 16	cfv 6546	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11215	. . . . . . 7 class ·
19	8, 17, 18	co 7427	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7427	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7427	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6546	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11551	. . . . . 6 class −
24	19, 22, 23	co 7427	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15865	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6546	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7427	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5188	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1569	1 wff log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))

This definition is used by:  lgamgulm2  27375  lgamf  27381  iprodgam  36634