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Definition df-lgam 27062
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 27059 . 2 class log Γ
2 vz . . 3 setvar 𝑧
3 cc 11153 . . . 4 class
4 cz 12613 . . . . 5 class
5 cn 12266 . . . . 5 class
64, 5cdif 3948 . . . 4 class (ℤ ∖ ℕ)
73, 6cdif 3948 . . 3 class (ℂ ∖ (ℤ ∖ ℕ))
82cv 1539 . . . . . . 7 class 𝑧
9 vm . . . . . . . . . . 11 setvar 𝑚
109cv 1539 . . . . . . . . . 10 class 𝑚
11 c1 11156 . . . . . . . . . 10 class 1
12 caddc 11158 . . . . . . . . . 10 class +
1310, 11, 12co 7431 . . . . . . . . 9 class (𝑚 + 1)
14 cdiv 11920 . . . . . . . . 9 class /
1513, 10, 14co 7431 . . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16 clog 26596 . . . . . . . 8 class log
1715, 16cfv 6561 . . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18 cmul 11160 . . . . . . 7 class ·
198, 17, 18co 7431 . . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
208, 10, 14co 7431 . . . . . . . 8 class (𝑧 / 𝑚)
2120, 11, 12co 7431 . . . . . . 7 class ((𝑧 / 𝑚) + 1)
2221, 16cfv 6561 . . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23 cmin 11492 . . . . . 6 class
2419, 22, 23co 7431 . . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
255, 24, 9csu 15722 . . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
268, 16cfv 6561 . . . 4 class (log‘𝑧)
2725, 26, 23co 7431 . . 3 class 𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
282, 7, 27cmpt 5225 . 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
291, 28wceq 1540 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  27079  lgamf  27085  iprodgam  35742
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