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Second partial derivative of $γ^r$ with respect to $τ$

Values for calculation

$ n $
$ \{π\in\mathbb{R}\mid π>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ π $
$ I $
$ J $
$ \{τ\in\mathbb{R}\mid τ>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ τ $
$ \text{Region} $

Calculation

Second partial derivative of $ γ^r $ with respect to $ τ $

$\text{if }\ \text{Region}= 2$
$\text{if }\ \text{Region}= 2$
Equations in LaTeX code
\text{if }\ \text{Region}= 2
$$γ^r_{ττ}=\sum_{i=1}^{43}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot\left(τ-0.5\right)^{J[i]-2}}$$
$γ^r_{ττ}=\sum_{i=1}^{43}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot\left(τ-0.5\right)^{J[i]-2}}$
Equations in LaTeX code
γ^r_{ττ}=\sum_{i=1}^{43}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot\left(τ-0.5\right)^{J[i]-2}}
$\text{else if }\ \text{Region}= 5$
$\text{else if }\ \text{Region}= 5$
Equations in LaTeX code
\text{else if }\ \text{Region}= 5
$$γ^r_{ττ}=\sum_{i=1}^{6}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot τ^{J[i]-2}}$$
$γ^r_{ττ}=\sum_{i=1}^{6}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot τ^{J[i]-2}}$
Equations in LaTeX code
γ^r_{ττ}=\sum_{i=1}^{6}{n[i]\cdot π^{I[i]}\cdot J[i]\cdot\left(J[i]-1\right)\cdot τ^{J[i]-2}}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$γ^r_{ττ}=0$$
$γ^r_{ττ}=0$
Equations in LaTeX code
γ^r_{ττ}=0
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