From bb6e87dc2036e69f16dc6dad5930e757f3f8c6e7 Mon Sep 17 00:00:00 2001
From: Anastasia Popova <apopova@caltech.edu>
Date: Wed, 14 Feb 2024 11:01:37 -0800
Subject: [PATCH] Gamma function update in docs

---
 docs/src/P3Scheme.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/docs/src/P3Scheme.md b/docs/src/P3Scheme.md
index 0c8ffbbe25..c53cbcb147 100644
--- a/docs/src/P3Scheme.md
+++ b/docs/src/P3Scheme.md
@@ -125,10 +125,10 @@ As a result ``q\_{ice}`` can be expressed as a sum of inclomplete gamma function
 |      condition(s)                            |    ``q_{ice} = \int \! m(D) N'(D) \mathrm{d}D``                                          |         gamma representation          |
 |:---------------------------------------------|:-----------------------------------------------------------------------------------------|:---------------------------------------------|
 | ``D < D_{th}``                               | ``\int_{0}^{D_{th}} \! \frac{\pi}{6} \rho_i \ D^3 N'(D) \mathrm{d}D``                    | ``\frac{\pi}{6} \rho_i N_0 \lambda \,^{-(\mu \, + 4)} (\Gamma \,(\mu \, + 4) - \Gamma \,(\mu \, + 4, \lambda \,D_{th}))``|
-| ``q_{rim} = 0`` and ``D > D_{th}``           | ``\int_{D_{th}}^{\infty} \! \alpha_{va} \ D^{\beta_{va}} N'(D) \mathrm{d}D``             | ``\alpha_{va} \ N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1) + \Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{th}) - (\mu \, + \beta_{va} \,)\Gamma \,(\mu \, + \beta_{va} \,))`` |
+| ``q_{rim} = 0`` and ``D > D_{th}``           | ``\int_{D_{th}}^{\infty} \! \alpha_{va} \ D^{\beta_{va}} N'(D) \mathrm{d}D``             | ``\alpha_{va} \ N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{th}))`` |
 | ``q_{rim} > 0`` and ``D_{gr} > D > D_{th}``  | ``\int_{D_{th}}^{D_{gr}} \! \alpha_{va} \ D^{\beta_{va}} N'(D) \mathrm{d}D``             | ``\alpha_{va} \ N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{th}) - \Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{gr}))`` |
 | ``q_{rim} > 0`` and ``D_{cr} > D > D_{gr}``  | ``\int_{D_{gr}}^{D_{cr}} \! \frac{\pi}{6} \rho_g \ D^3 N'(D) \mathrm{d}D``               | ``\frac{\pi}{6} \rho_g N_0 \lambda \,^{-(\mu \, + 4)} (\Gamma \,(\mu \, + 4, \lambda \,D_{gr}) - \Gamma \,(\mu \, + 4, \lambda \,D_{cr}))`` |
-| ``q_{rim} > 0`` and ``D > D_{cr}``           | ``\int_{D_{cr}}^{\infty} \! \frac{\alpha_{va}}{1-F_r} D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\frac{\alpha_{va}}{1-F_r} N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1) + \Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{cr}) - (\mu \, + \beta_{va} \,)\Gamma \,(\mu \, + \beta_{va} \,))``  |
+| ``q_{rim} > 0`` and ``D > D_{cr}``           | ``\int_{D_{cr}}^{\infty} \! \frac{\alpha_{va}}{1-F_r} D^{\beta_{va}} N'(D) \mathrm{d}D`` | ``\frac{\alpha_{va}}{1-F_r} N_0 \lambda \,^{-(\mu \, + \beta_{va} \, + 1)} (\Gamma \,(\mu \, + \beta_{va} \, + 1, \lambda \,D_{cr}))``  |
 
 where ``\Gamma \,(a, z) = \int_{z}^{\infty} \! t^{a - 1} e^{-t} \mathrm{d}D``
   and ``\Gamma \,(a) = \Gamma \,(a, 0)`` for simplicity.