Chapter 14.2, Problem 43P

In Chapter 12, equations (5.1) and $(5.2),$ we expanded the function $\varphi (x,h)$ in a series of powers of $h$.

Use Theorem III (see instructions for Problems 34 to 42 above) to show that the series for $\varphi (x,h)$ converges for $\left|h\right|<1$ and $-1\le x\le 1.$ Here $h$ is the variable and $x$ is a parameter; you should find the (complex) value of $h$ which makes $\Phi$ infinite, and show that the absolute value of this complex number is 1 (independent of $x$ when $x^2\le 1$ ). This proves that the series for real $h$ converges for $\left|h\right|<1$.