Definition df-lgam 27346

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 27343	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11198	. . . 4 class ℂ
4		cz 12693	. . . . 5 class ℤ
5		cn 12335	. . . . 5 class ℕ
6	4, 5	cdif 3896	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3896	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1569	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1569	. . . . . . . . . 10 class 𝑚
11		c1 11201	. . . . . . . . . 10 class 1
12		caddc 11203	. . . . . . . . . 10 class +
13	10, 11, 12	co 7420	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11973	. . . . . . . . 9 class /
15	13, 10, 14	co 7420	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26882	. . . . . . . 8 class log
17	15, 16	cfv 6538	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11205	. . . . . . 7 class ·
19	8, 17, 18	co 7420	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7420	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7420	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6538	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11541	. . . . . 6 class −
24	19, 22, 23	co 7420	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15853	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6538	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7420	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5186	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1570	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  27363  lgamf  27369  iprodgam  36507