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Definition df-lgam 25523
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 25520 . 2 class log Γ
2 vz . . 3 setvar 𝑧
3 cc 10523 . . . 4 class
4 cz 11969 . . . . 5 class
5 cn 11626 . . . . 5 class
64, 5cdif 3930 . . . 4 class (ℤ ∖ ℕ)
73, 6cdif 3930 . . 3 class (ℂ ∖ (ℤ ∖ ℕ))
82cv 1527 . . . . . . 7 class 𝑧
9 vm . . . . . . . . . . 11 setvar 𝑚
109cv 1527 . . . . . . . . . 10 class 𝑚
11 c1 10526 . . . . . . . . . 10 class 1
12 caddc 10528 . . . . . . . . . 10 class +
1310, 11, 12co 7145 . . . . . . . . 9 class (𝑚 + 1)
14 cdiv 11285 . . . . . . . . 9 class /
1513, 10, 14co 7145 . . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16 clog 25065 . . . . . . . 8 class log
1715, 16cfv 6348 . . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18 cmul 10530 . . . . . . 7 class ·
198, 17, 18co 7145 . . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
208, 10, 14co 7145 . . . . . . . 8 class (𝑧 / 𝑚)
2120, 11, 12co 7145 . . . . . . 7 class ((𝑧 / 𝑚) + 1)
2221, 16cfv 6348 . . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23 cmin 10858 . . . . . 6 class
2419, 22, 23co 7145 . . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
255, 24, 9csu 15030 . . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
268, 16cfv 6348 . . . 4 class (log‘𝑧)
2725, 26, 23co 7145 . . 3 class 𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
282, 7, 27cmpt 5137 . 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
291, 28wceq 1528 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  25540  lgamf  25546  iprodgam  32871
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