Definition df-lgam 26983

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 26980	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11022	. . . 4 class ℂ
4		cz 12486	. . . . 5 class ℤ
5		cn 12143	. . . . 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 1540	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1540	. . . . . . . . . 10 class 𝑚
11		c1 11025	. . . . . . . . . 10 class 1
12		caddc 11027	. . . . . . . . . 10 class +
13	10, 11, 12	co 7356	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11792	. . . . . . . . 9 class /
15	13, 10, 14	co 7356	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26517	. . . . . . . 8 class log
17	15, 16	cfv 6490	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11029	. . . . . . 7 class ·
19	8, 17, 18	co 7356	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7356	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7356	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6490	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11362	. . . . . 6 class −
24	19, 22, 23	co 7356	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15607	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6490	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7356	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5177	. 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  27000  lgamf  27006  iprodgam  35885