Definition df-lgam 26450

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 26447	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11090	. . . 4 class ℂ
4		cz 12540	. . . . 5 class ℤ
5		cn 12194	. . . . 5 class ℕ
6	4, 5	cdif 3941	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3941	. . 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 11093	. . . . . . . . . 10 class 1
12		caddc 11095	. . . . . . . . . 10 class +
13	10, 11, 12	co 7393	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11853	. . . . . . . . 9 class /
15	13, 10, 14	co 7393	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25992	. . . . . . . 8 class log
17	15, 16	cfv 6532	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11097	. . . . . . 7 class ·
19	8, 17, 18	co 7393	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7393	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7393	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6532	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11426	. . . . . 6 class −
24	19, 22, 23	co 7393	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15614	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6532	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7393	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5224	. 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  26467  lgamf  26473  iprodgam  34540