Definition df-lgam 27080

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 27077	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11071	. . . 4 class ℂ
4		cz 12568	. . . . 5 class ℤ
5		cn 12210	. . . . 5 class ℕ
6	4, 5	cdif 3901	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3901	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1559	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1559	. . . . . . . . . 10 class 𝑚
11		c1 11074	. . . . . . . . . 10 class 1
12		caddc 11076	. . . . . . . . . 10 class +
13	10, 11, 12	co 7396	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11844	. . . . . . . . 9 class /
15	13, 10, 14	co 7396	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26616	. . . . . . . 8 class log
17	15, 16	cfv 6521	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11078	. . . . . . 7 class ·
19	8, 17, 18	co 7396	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7396	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7396	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6521	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11414	. . . . . 6 class −
24	19, 22, 23	co 7396	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15713	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6521	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7396	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5181	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1560	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  27097  lgamf  27103  iprodgam  36089