Definition df-lgam 27183

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 27180	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11093	. . . 4 class ℂ
4		cz 12586	. . . . 5 class ℤ
5		cn 12228	. . . . 5 class ℕ
6	4, 5	cdif 3902	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3902	. . 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 11096	. . . . . . . . . 10 class 1
12		caddc 11098	. . . . . . . . . 10 class +
13	10, 11, 12	co 7410	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11866	. . . . . . . . 9 class /
15	13, 10, 14	co 7410	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26719	. . . . . . . 8 class log
17	15, 16	cfv 6536	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11100	. . . . . . 7 class ·
19	8, 17, 18	co 7410	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7410	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7410	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6536	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11436	. . . . . 6 class −
24	19, 22, 23	co 7410	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15733	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6536	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7410	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5192	. 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 referenced by:  lgamgulm2  27200  lgamf  27206  iprodgam  36234