Definition df-lgam 26405

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 26402	. 2 class log Γ
2		vz	. . 3 setvar 𝑧
3		cc 11058	. . . 4 class ℂ
4		cz 12508	. . . . 5 class ℤ
5		cn 12162	. . . . 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 1540	. . . . . . 7 class 𝑧
9		vm	. . . . . . . . . . 11 setvar 𝑚
10	9	cv 1540	. . . . . . . . . 10 class 𝑚
11		c1 11061	. . . . . . . . . 10 class 1
12		caddc 11063	. . . . . . . . . 10 class +
13	10, 11, 12	co 7362	. . . . . . . . 9 class (𝑚 + 1)
14		cdiv 11821	. . . . . . . . 9 class /
15	13, 10, 14	co 7362	. . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16		clog 25947	. . . . . . . 8 class log
17	15, 16	cfv 6501	. . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18		cmul 11065	. . . . . . 7 class ·
19	8, 17, 18	co 7362	. . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
20	8, 10, 14	co 7362	. . . . . . . 8 class (𝑧 / 𝑚)
21	20, 11, 12	co 7362	. . . . . . 7 class ((𝑧 / 𝑚) + 1)
22	21, 16	cfv 6501	. . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23		cmin 11394	. . . . . 6 class −
24	19, 22, 23	co 7362	. . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
25	5, 24, 9	csu 15582	. . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
26	8, 16	cfv 6501	. . . 4 class (log‘𝑧)
27	25, 26, 23	co 7362	. . 3 class (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
28	2, 7, 27	cmpt 5193	. 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  26422  lgamf  26428  iprodgam  34401