Definition df-lgam 27236

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 27233	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11115	. . . 4 class ℂ
4		cz 12608	. . . . 5 class ℤ
5		cn 12250	. . . . 5 class ℕ
6	4, 5	cdif 3903	. . . 4 class (ℤ ∖ ℕ)
7	3, 6	cdif 3903	. . 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 11118	. . . . . . . . . 10 class 1
12		caddc 11120	. . . . . . . . . 10 class +
13	10, 11, 12	co 7419	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11888	. . . . . . . . 9 class /
15	13, 10, 14	co 7419	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 26772	. . . . . . . 8 class log
17	15, 16	cfv 6540	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11122	. . . . . . 7 class ·
19	8, 17, 18	co 7419	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7419	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7419	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6540	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11458	. . . . . 6 class −
24	19, 22, 23	co 7419	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15763	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6540	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7419	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5194	. 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  27253  lgamf  27259  iprodgam  36273