The Special Function {E\left(k\right)}

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1. Differential Equation[-]

The function {E\left(k\right)} satisfies the differential equation with initial values y \left( 0 \right) =\frac{\pi }{2} and y'' \left( 0 \right) =-\frac{\pi }{4}.

2. Plot[+]

3. Derivative in Terms of Lower-Order Derivatives[+]

4. Expansion at 0[+]

5. Local Expansions at Singularities and at Infinity[-]

The differential equation above has 2 non-zero finite singular pointsA complex numberĀ z_0 is a singular point (or singularity of a linear ordinary differential equation with polynomial coefficients, if the leading coefficient of the equation vanishes atĀ z_0..

5.1. Expansion at -1[+]

5.2. Expansion at 1[+]

5.3. Expansion at \infty [+]

6. Hypergeometric Representation[+]

7. Chebyshev Expansion over [-1,1][+]