Definition df-lgam 26985

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 26982	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11024	. . . 4 class ℂ
4		cz 12488	. . . . 5 class ℤ
5		cn 12145	. . . . 5 class ℕ
6	4, 5	cdif 3898	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3898	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1540	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1540	. . . . . . . . . 10 class 𝑚
11		c1 11027	. . . . . . . . . 10 class 1
12		caddc 11029	. . . . . . . . . 10 class +
13	10, 11, 12	co 7358	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11794	. . . . . . . . 9 class /
15	13, 10, 14	co 7358	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26519	. . . . . . . 8 class log
17	15, 16	cfv 6492	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11031	. . . . . . 7 class ·
19	8, 17, 18	co 7358	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7358	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7358	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6492	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11364	. . . . . 6 class −
24	19, 22, 23	co 7358	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15609	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6492	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7358	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5179	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1541	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  27002  lgamf  27008  iprodgam  35936