Notes on Whittaker & Watson, Chapter VII, part 1

Whittaker & Watson, A Course of Modern Analysis. Chapter VII, The Expansion of Function in Infinite Series.

7.1 Darboux’s formula calculates $f(z)-f(a)$ via an $n$th order polynomial $\phi(t)$ using the values of the function at $a$ and its derivatives up to $n$th order at $a$ and $z$ $$ f(z)-f(a) = \frac{1}{\phi^{(n)}(0)}\sum_{m=1}^n(-1)^{m-1}(z-a)^m(\phi^{(n-m)}(1)f^{(m)}(z)-\phi^{(n-m)}(0)f^{(m)}(a)) + R_n(z), $$ where the remainder $$ R_n(z) = (-1)^{n}(z-a)^{n+1}\int_0^1\phi(t)f^{(n+1)}(a+t(z-a))dt. $$ The Taylor series can be seen as a specific case of Darboux’s formula with $\phi(t)=(t-1)^n$.

7.2 Bernoullian numbers $B_n$ as coefficients of the Maclaurin series for $\frac{z}{2}\cot\frac{z}{2}$ $$ B_n=-\left.\frac{d^{2n}}{dz^{2n}}\right|_{z\to0}\frac{z}{2}\cot\frac{z}{2} $$ and in its integral form $$ B_n=4n\int_0^\infty\frac{t^{2n-1}dt}{e^{2\pi t-1}} $$ which can be obtained by differentiating the integral $\int_0^\infty\frac{\sin{px}\ dx}{e^{\pi x}-1} = \frac{i}{2}\cot{ip}-\frac{1}{2p}$ with respect to $p$ which has the series expansion with coefficients expressed by $B_n$.

[Note that the convention followed in Whittaker & Watson omits the odd Bernoulli numbers.]

Bernoullian polynomial $\phi_n(z)$ of order $n$ is defined by the generating function $t\frac{e^{zt}-1}{e^t-1}$, such that $$ t\frac{e^{zt}-1}{e^t-1}=\sum_{n=1}^\infty\phi_n(z)\frac{t^n}{n!}. $$ It satisfies the following difference equation: $$ nz^{n-1}=\phi_n(z+1)-\phi_n(z). $$ By expanding $e^{zt}-1$ and $\frac{t}{e^t-1}=\frac{t}{2i}\cot\frac{t}{2i}$, we have the explicit expression

$$ \phi_n(z)=z^n-\frac{n}{2}z^{n-1}+{n \choose 2}B_1z^{n-1}-{n \choose 4}B_2z^{n-4}+{n \choose 6}B_3z^{n-6}-\cdots. $$ 7.21 Using the Bernoullian polynomial in Darboux’s formula, we have the Euler-Maclaurin sum formula, which calculates the sum of a function $\sum_{m=0}^rF(a+m\omega)$ by the integral $\int_a^{a+r\omega}F(x)dx$ corrected by the derivatives of $F$ at $a+m\omega$ with coefficients involving $B_m$.

7.3 Reverting the Taylor series $\phi(z)-b$ for an analytic function $\phi$ at $\phi(a)=b$,

$$ z-a = \frac{1}{\phi'(a)}(\phi(z)-b) - \frac{1}{2}\frac{\phi''(a)}{\phi'(a)^3} (\phi(z)-b)^2 + \cdots, $$ Bürmann’s theorem gives the coefficients of the expansion of $f(z) - f(a)$ as series of $(\phi(z)-b)^m$.

7.31 Teixeira’s extended form of Bürmann’s theorem expands $f(z)$ as a series of $\theta(z)^m$ on a ring-shaped region, instead of $z^m$ as in the Laurent series.

7.32 With $\theta(z)=\frac{(z-a)}{\phi(z)}$ in which $\phi$ is analytic in 7.31, Lagrange’s theorem provides the series expansion

$$ f(\zeta) = f(a)+\sum_{n=1}^\infty\frac{t^n}{n!}\frac{d^{n-1}}{da^{n-1}}[f'(a)\phi(a)^n] $$ where $t=\theta(\zeta)$.