Definition df-lgam 27256

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 27253	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11123	. . . 4 class ℂ
4		cz 12616	. . . . 5 class ℤ
5		cn 12258	. . . . 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 11126	. . . . . . . . . 10 class 1
12		caddc 11128	. . . . . . . . . 10 class +
13	10, 11, 12	co 7414	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11896	. . . . . . . . 9 class /
15	13, 10, 14	co 7414	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26792	. . . . . . . 8 class log
17	15, 16	cfv 6533	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11130	. . . . . . 7 class ·
19	8, 17, 18	co 7414	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7414	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7414	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6533	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11466	. . . . . 6 class −
24	19, 22, 23	co 7414	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15774	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6533	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7414	. . 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  27273  lgamf  27279  iprodgam  36322