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Definition df-lgam 25512
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
StepHypRef Expression
1 clgam 25509 . 2 class log Γ
2 vz . . 3 setvar 𝑧
3 cc 10527 . . . 4 class
4 cz 11973 . . . . 5 class
5 cn 11630 . . . . 5 class
64, 5cdif 3936 . . . 4 class (ℤ ∖ ℕ)
73, 6cdif 3936 . . 3 class (ℂ ∖ (ℤ ∖ ℕ))
82cv 1529 . . . . . . 7 class 𝑧
9 vm . . . . . . . . . . 11 setvar 𝑚
109cv 1529 . . . . . . . . . 10 class 𝑚
11 c1 10530 . . . . . . . . . 10 class 1
12 caddc 10532 . . . . . . . . . 10 class +
1310, 11, 12co 7151 . . . . . . . . 9 class (𝑚 + 1)
14 cdiv 11289 . . . . . . . . 9 class /
1513, 10, 14co 7151 . . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16 clog 25053 . . . . . . . 8 class log
1715, 16cfv 6351 . . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18 cmul 10534 . . . . . . 7 class ·
198, 17, 18co 7151 . . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
208, 10, 14co 7151 . . . . . . . 8 class (𝑧 / 𝑚)
2120, 11, 12co 7151 . . . . . . 7 class ((𝑧 / 𝑚) + 1)
2221, 16cfv 6351 . . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23 cmin 10862 . . . . . 6 class
2419, 22, 23co 7151 . . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
255, 24, 9csu 15035 . . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
268, 16cfv 6351 . . . 4 class (log‘𝑧)
2725, 26, 23co 7151 . . 3 class 𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
282, 7, 27cmpt 5142 . 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
291, 28wceq 1530 1 wff log Γ = (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
Colors of variables: wff setvar class
This definition is referenced by:  lgamgulm2  25529  lgamf  25535  iprodgam  32860
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