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Definition df-lgam 25097
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 25094 . 2 class log Γ
2 vz . . 3 setvar 𝑧
3 cc 10222 . . . 4 class
4 cz 11666 . . . . 5 class
5 cn 11312 . . . . 5 class
64, 5cdif 3766 . . . 4 class (ℤ ∖ ℕ)
73, 6cdif 3766 . . 3 class (ℂ ∖ (ℤ ∖ ℕ))
82cv 1652 . . . . . . 7 class 𝑧
9 vm . . . . . . . . . . 11 setvar 𝑚
109cv 1652 . . . . . . . . . 10 class 𝑚
11 c1 10225 . . . . . . . . . 10 class 1
12 caddc 10227 . . . . . . . . . 10 class +
1310, 11, 12co 6878 . . . . . . . . 9 class (𝑚 + 1)
14 cdiv 10976 . . . . . . . . 9 class /
1513, 10, 14co 6878 . . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16 clog 24642 . . . . . . . 8 class log
1715, 16cfv 6101 . . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18 cmul 10229 . . . . . . 7 class ·
198, 17, 18co 6878 . . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
208, 10, 14co 6878 . . . . . . . 8 class (𝑧 / 𝑚)
2120, 11, 12co 6878 . . . . . . 7 class ((𝑧 / 𝑚) + 1)
2221, 16cfv 6101 . . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23 cmin 10556 . . . . . 6 class
2419, 22, 23co 6878 . . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
255, 24, 9csu 14757 . . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
268, 16cfv 6101 . . . 4 class (log‘𝑧)
2725, 26, 23co 6878 . . 3 class 𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
282, 7, 27cmpt 4922 . 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
291, 28wceq 1653 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  25114  lgamf  25120  iprodgam  32142
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