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CritPt / Challenge_5_main / Let α∈[0,1] be a real parameter. Define the function f(n,α)…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
import sympy as sp
alpha = sp.symbols('alpha', real=True)
def answer(alpha):
r"""
Return the expression of the derivative $g(\alpha)$ in Sympy format
Inputs
----------
alpha: sympy.Symbol, real parameter, $\alpha \in [0, 1]$
Outputs
----------
g_alpha: sympy.Expr, $g(\alpha)$ for $\alpha \in [0, 1]$.
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
g_alpha = ...
# ---------------------------------------------------------------
return g_alphaproblem description
# Problem setup:
Let be a real parameter. Define the function for complex by
\begin{equation}
f(n, \alpha) = (1 + \alpha)^{n - 1} \, {}_2F_1\left( \frac{1 - n}{2}, 1 - \frac{n}{2}; 2; \left( \frac{2\sqrt{\alpha}}{1 + \alpha} \right)^2 \right),
\end{equation}
where denotes the Gaussian hypergeometric function.
# Main problem:
Evaluate the derivative
\begin{equation}
g(\alpha) = \left. \frac{\partial}{\partial n} f(n, \alpha) \right|_{n = 0}
\end{equation}
for .
Plain-text mathematical notation (without MathML)
# Problem setup:
Let α∈[0,1] be a real parameter. Define the function f(n,α) for complex n by
\begin{equation}
f(n, \alpha) = (1 + \alpha)^{n - 1} \, {}_2F_1\left( \frac{1 - n}{2}, 1 - \frac{n}{2}; 2; \left( \frac{2\sqrt{\alpha}}{1 + \alpha} \right)^2 \right),
\end{equation}
where ₂F₁(a,b;c;z) denotes the Gaussian hypergeometric function.
# Main problem:
Evaluate the derivative
\begin{equation}
g(\alpha) = \left. \frac{\partial}{\partial n} f(n, \alpha) \right|_{n = 0}
\end{equation}
for α∈[0,1].Original LaTeX notation
# Problem setup:
Let $\alpha \in [0, 1]$ be a real parameter. Define the function $f(n, \alpha)$ for complex $n$ by
\begin{equation}
f(n, \alpha) = (1 + \alpha)^{n - 1} \, {}_2F_1\left( \frac{1 - n}{2}, 1 - \frac{n}{2}; 2; \left( \frac{2\sqrt{\alpha}}{1 + \alpha} \right)^2 \right),
\end{equation}
where ${}_2F_1(a, b; c; z)$ denotes the Gaussian hypergeometric function.
# Main problem:
Evaluate the derivative
\begin{equation}
g(\alpha) = \left. \frac{\partial}{\partial n} f(n, \alpha) \right|_{n = 0}
\end{equation}
for $\alpha \in [0, 1]$.Discussion
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initial import