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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.

problem description

# Problem setup: Let α∈[0,1]\alpha \in [0, 1] be a real parameter. Define the function f(n,α)f(n, \alpha) for complex nn 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){}_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 α∈[0,1]\alpha \in [0, 1].
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 $\alpha \in [0, 1]$.

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_alpha

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Source and history

upstream_id / Original problem ID
Challenge_5_main
task_id
0a033e35-140a-5604-84a6-d78e8af0e7a1
task_key
train--Challenge~5f5~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import