Definition df-lgam 27148

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 27145	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11097	. . . 4 class ℂ
4		cz 12590	. . . . 5 class ℤ
5		cn 12232	. . . . 5 class ℕ
6	4, 5	cdif 3910	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3910	. . 3 class (ℂ ∖ (ℤ ∖ ℕ))
8	2	cv 1566	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1566	. . . . . . . . . 10 class 𝑚
11		c1 11100	. . . . . . . . . 10 class 1
12		caddc 11102	. . . . . . . . . 10 class +
13	10, 11, 12	co 7411	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11870	. . . . . . . . 9 class /
15	13, 10, 14	co 7411	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26684	. . . . . . . 8 class log
17	15, 16	cfv 6537	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11104	. . . . . . 7 class ·
19	8, 17, 18	co 7411	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7411	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7411	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6537	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11440	. . . . . 6 class −
24	19, 22, 23	co 7411	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15736	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6537	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7411	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5196	. 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
29	1, 28	wceq 1567	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  27165  lgamf  27171  iprodgam  36132