Hint for Assignment 5

This is a hint for problem 12, p. 280. First, use the definitions of H_ν⁽¹⁾(z) and H_ν⁽²⁾(z) for fractional order ν, with 0 ≤ arg(z) < 2π, to show that

H_ν⁽¹⁾(z) = i csc(π ν)( e^{−iπ ν} J_ν(z) − J_−ν(z))

and

H_ν⁽²⁾(z) = −i csc(π ν)( e^{iπ ν} J_ν(z) − J_−ν(z))

Next, use these formulas to show that

H_ν⁽¹⁾(−z) = − e^{−iπ ν} H_ν⁽²⁾(z)

and

H_ν⁽²⁾(−z) = − e ^{iπ ν} H_ν⁽¹⁾(z)

both hold, provided 0 ≤ arg(z) < π. In particular, both formulas will hold if z = x is real and positive. Finally, all the functions above are analytic in ν. By letting ν → 0, we have that

H₀⁽¹⁾(−z) = − H₀⁽²⁾(z)

and

H₀⁽²⁾(−z) = − H₀⁽¹⁾(z).

Judiciously using these in the problem will give the solution.