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Definition df-lgam 26450
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 26447 . 2 class log Γ
2 vz . . 3 setvar 𝑧
3 cc 11090 . . . 4 class
4 cz 12540 . . . . 5 class
5 cn 12194 . . . . 5 class
64, 5cdif 3941 . . . 4 class (ℤ ∖ ℕ)
73, 6cdif 3941 . . 3 class (ℂ ∖ (ℤ ∖ ℕ))
82cv 1540 . . . . . . 7 class 𝑧
9 vm . . . . . . . . . . 11 setvar 𝑚
109cv 1540 . . . . . . . . . 10 class 𝑚
11 c1 11093 . . . . . . . . . 10 class 1
12 caddc 11095 . . . . . . . . . 10 class +
1310, 11, 12co 7393 . . . . . . . . 9 class (𝑚 + 1)
14 cdiv 11853 . . . . . . . . 9 class /
1513, 10, 14co 7393 . . . . . . . 8 class ((𝑚 + 1) / 𝑚)
16 clog 25992 . . . . . . . 8 class log
1715, 16cfv 6532 . . . . . . 7 class (log‘((𝑚 + 1) / 𝑚))
18 cmul 11097 . . . . . . 7 class ·
198, 17, 18co 7393 . . . . . 6 class (𝑧 · (log‘((𝑚 + 1) / 𝑚)))
208, 10, 14co 7393 . . . . . . . 8 class (𝑧 / 𝑚)
2120, 11, 12co 7393 . . . . . . 7 class ((𝑧 / 𝑚) + 1)
2221, 16cfv 6532 . . . . . 6 class (log‘((𝑧 / 𝑚) + 1))
23 cmin 11426 . . . . . 6 class
2419, 22, 23co 7393 . . . . 5 class ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
255, 24, 9csu 15614 . . . 4 class Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1)))
268, 16cfv 6532 . . . 4 class (log‘𝑧)
2725, 26, 23co 7393 . . 3 class 𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧))
282, 7, 27cmpt 5224 . 2 class (𝑧 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↦ (Σ𝑚 ∈ ℕ ((𝑧 · (log‘((𝑚 + 1) / 𝑚))) − (log‘((𝑧 / 𝑚) + 1))) − (log‘𝑧)))
291, 28wceq 1541 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  26467  lgamf  26473  iprodgam  34540
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