Notes on Whittaker & Watson, Chapter XI, part 3

Whittaker & Watson, A Course of Modern Analysis. Chapter XI, Integral Equations.

11.3 Introducing Fredholm’s integral equation of the first kind $$ f(x)=\lambda\int_a^bK(x,\xi)\phi(\xi)d\xi $$ with $\phi(\cdot)$ unknown, and the integral equations with variable upper limits $$ f(x)=\lambda\int_a^xK(x,\xi)\phi(\xi)d\xi\newline \phi(x) = f(x)+\lambda\int_a^xK(x,\xi)\phi(\xi)d\xi. $$ 11.31. The integral equation of the first kind with variable upper limit is also known as Volterra’s equation.

When $\frac{\partial K}{\partial x}$ exists and is continuous, we can differentiate the equation to get $$ f'(x) = \lambda K(x,x)\phi(x) + \lambda\int_a^x\frac{\partial K}{\partial x}\phi(\xi)d\xi $$ which is an equation of the second kind. The solution to this equation also solves Volterra’s equation when $f(a)=0$.

11.4 The Liouville-Neumann method of successive substitution. Recursively substituting $\phi(\xi)$ in the expression on the RHS by the equation, we obtain the series $$ S(x) = f(x)+\lambda\int_a^bK(x,\xi)f(\xi)d\xi+\sum_{m=2}^\infty\lambda^m\int_a^bK(x,\xi_1)\int_a^bK(\xi_1,\xi_2)\newline\cdots\int_a^bK(\xi_{m-1},\xi_m)f(\xi_m)d\xi_m\cdots d\xi_1 $$ which converges uniformly for $|\lambda|<M^{-1}(b-a)^{-1}$.

For reciprocal (§11.22) $k(x,\xi;\lambda)=-K(x,\xi) + \lambda\int_a^bk(x,\xi_1;\lambda)K(\xi,\xi_1)d\xi_1$, the substitution yields $$ k(x,\xi;\lambda)=-K(x,-\xi) -\sum_{m=2}^\infty\lambda^{m-1}\int_a^bK(x,\xi_1)\int_a^bK(\xi_1,\xi_2)\newline\cdots\int_a^bK(\xi_{m-1},\xi)d\xi_{m-1}\cdots d\xi_1. $$ Thus the series solution $S(x)$ above can be rewritten as $$ S(x) = f(x) -\lambda\int_a^bk(x,\xi;\lambda)f(\xi)d\xi. $$ With $K_1(x,\xi)\equiv K(x,\xi)$ and $K_{n+1}(x,\xi) \equiv \int_a^b K(x,\xi')K_n(\xi', \xi)d\xi'$, we have $$ k(x,\xi;\lambda) = -\sum_{m=0}^\infty\lambda^mK_{m+1}(x,\xi) $$ and $$ \int_a^bK_m(x,\xi')K_n(\xi',\xi)d\xi'=K_{m+n}(x,\xi). $$ 11.5. For symmetric kernel $K(x,y)=K(y,x)$ and by induction it can be proved that $K_n(x,y)=K_n(y,x)$.

11.51 Schmidt’s theorem: for symmetric kernel $K$, the determinant $D(\lambda)=0$ has at least one root.

The existence of at least one root can be proved from the existence of singularities in the derivative of $\log D(\lambda)$. Consider its series expansion $$ -\frac{1}{D(\lambda)}\frac{dD(\lambda)}{d\lambda} = \sum_{n=1}^\infty U_n\lambda^{n-1} $$ where $U_n=\int_a^b K(x,x)dx$. From $\int\int(\mu K_{n+1}(x,\xi)+\mu K_{n-1}(x,\xi))^2dxd\xi\ge 0, \forall \mu\in\mathbb{R}$, we have $U_{2n+2}U_{2n-2}\ge U_{2n}^2$. Hence $\frac{U_{2n+2}}{U_{2n}}\ge\nu^n$ with $\nu\equiv\frac{U_4}{U_2}$. The series therefore does not converge for $|\lambda|\ge\nu^{-1}$, indicating the existence of singularities. Since $D(\lambda)$ is entire, the singularity must be due to $\frac{1}{D(\lambda)}$.

11.6 A set of orthonormal functions $\{\phi_1(x), \phi_2(x), \cdots\}$ with inner product $$ \int_a^b\phi_m(x)\phi_n(x)dx=\delta_{mn}. $$ 11.61 For homogeneous equation $$ \phi(x)=\lambda_0\int_a^b\phi(\xi)K(x,\xi)d\xi $$ where $\lambda_0$ is a real characteristic number (§11.23) of $K$, an orthonormal basis $\{\phi_1(x), \phi_2(x), \cdots\phi_n(x)\}$ can be formed from any $n$ of its linear independent solutions. The multiplicity is bounded by the characteristic number $\lambda_0$ $$ n \leq \lambda_0^2\int_a^b\int_a^bK(x,y)^2dx,dy $$ [Alternatively, this gives a bound on the spectral radius of the integral operator, with $\{\frac{1}{\lambda}\}$ being the eigenvalues. The spectral radius is determined by the operator norm (Gelfand’s formula). The operator norm of the integral transform is bounded by the $L_2$-norm of the kernel $K$, which can be proved by Cauchy-Schwarz inequality.]

The characteristic [eigen-]functions corresponding to different $\lambda$-s are orthogonal $$ \int_a^b\phi^{(0)}(x)\phi^{(1)}(x)dx = 0 $$ where $\phi^{(i)}$ is a characteristic function for $\lambda_i$ and $\lambda_0\neq\lambda_1$.

Characteristic numbers of symmetric kernels are real.

11.7 With characteristic functions $\phi_1, \phi_2, \phi_3,\cdots$ corresponding to characteristic numbers $\lambda_1, \lambda_2, \lambda_3, \cdots$ in which $\lambda_i$ and $\lambda_j$ are not necessarily distinct, the kernel can be expressed $$ K(x,y)=\sum_{n=1}^\infty \frac{\phi_n(x)\phi_n(y)}{\lambda_n}. $$ 11.71 Thus Fredholm’s equation $$ \Phi(x)=f(x)+\lambda\int_a^bK(x,\xi)\Phi(\xi)d\xi $$ for symmetric kernel $K(\cdot,\cdot)$ can be solved by eigenfunction expansion $\Phi=\sum\frac{b_n\lambda_n}{\lambda_n-\lambda}\phi_n$ where $b_n$ are the coefficients for the expansion $f=\sum b_n\phi_n$.

11.8 Abel’s integral equation $$ f(x)=\int_a^x\frac{u(\xi)}{(x-\xi)^\mu}d\xi\quad\quad(0<\mu<1,a\leq x\leq b) $$ with $f(a)=0$.

Consider the integral $$ \begin{align*} \int_a^z\frac{f(x)}{(z-x)^{1-\mu}}dx&=\int_a^zdx\int_a^x\frac{u(\xi)}{(x-\xi)^\mu(z-x)^{1-\mu}}d\xi\newline &= \int_a^xu(\xi)d\xi\int_\xi^z\frac{dx}{(x-\xi)^\mu(z-x)^{1-\mu}}\newline &=\frac{\pi}{\sin\mu\pi}\int_a^xu(\xi)d\xi. \end{align*} $$ Hence $$ u(z)=\frac{\sin\mu\pi}{\pi}\frac{d}{dz}\int_a^z\frac{f(x)}{(z-x)^{1-\mu}}. $$ 11.81 Schlömilch’s integral equation $$ f(x)=\frac{2}{\pi}\int_0^{\pi/2}\phi(x\sin\theta)d\theta\quad\quad(-\pi\le x\le\pi). $$ has solution $$ \phi(x) = f(0)+x\int_0^{\pi/2}f'(x\sin\psi)d\psi. $$