Fuchsian differential equation

A Fuchsian differential equation has only regular singular points, including possibly the point at infinity.

An accessory parameter is a coefficient not determined by the singular points and their indicial exponents. A second-order Fuchsian differential equation with exactly three regular singular points has no such independent parameter.

A regular singular point is a singularity at which solutions of a linear differential equation have at worst controlled power and logarithmic behavior after a Frobenius reduction.

A regular singular point with a full set of analytic local solutions is an apparent singularity of the differential equation. Singular normalized coefficients do not by themselves force singular solutions. For example, $x{y}^{\prime \prime }-y^{\prime }=0$ has normalized coefficient $-1/x$, but its fundamental solutions $1,x^2$ are both analytic through zero. Their Wronskian is $2x$, which vanishes at the singular point while being nonzero on each adjacent ordinary interval. The nonvanishing-Wronskian criterion for a fundamental set applies on intervals where the normalized equation has regular coefficients.

For $x{y}^{\prime \prime }+a(x)y=0$ with analytic coefficient and a continuous solution having $a(0)y(0)\ne 0$, the equation gives $y^{\prime \prime }\sim -a(0)y(0)/x$. Integrating towards the endpoint gives the displayed logarithmic divergence of the derivative. A bounded solution value therefore does not imply a finite derivative at a regular singular point.

Near a regular singular point $x_0$, the Frobenius method seeksThe lowest power gives the indicial equation for $r$, and the remaining powers give a coefficient recurrence.

For $a>0$, $b\ne 0$, and a positive integer $\beta$, consider $x^2{y}^{\prime \prime }+x{y}^{\prime }+(-a^2+b{x}^{\beta })y=0$ near $x=0$. Its indicial roots are $a,-a$. The first Frobenius solution, normalized to leading coefficient one, is $y_{+}=x^a{\sum }_{k\ge 0}{c}_{k\beta }{x}^{k\beta }$, with $c_{k\beta }=-b{c}_{(k-1)\beta }/[k\beta (k\beta +2a)]$. If $N=2a$ is a positive integer, a second pure Frobenius solution with leading coefficient one would requireBefore $n=N$, induction makes $d_n=0$ for indices not divisible by $\beta$. If $\beta$ divides $N$, the coefficient $d_{N-\beta }$ is nonzero, and the exceptional equation $0=-b{d}_{N-\beta }$ is impossible. If $\beta$ does not divide $N$, the right-hand side is zero and $d_N$ is free. Thus, in the Frobenius case classification, a positive integer gap gives Case IIIa exactly when $\beta$ divides $2a$, and Case IIIb otherwise.

The coefficient of the necessary logarithm can also be calculated. Let $L=x^2{d}^2/d{x}^2+xd/dx-a^2+b{x}^{\beta }$. The product rule gives $L(y_{+}\log x)=2x{y}_{+}^{\prime }$. Therefore a second solution $x^{-a}\sum d_n{x}^n+K{y}_{+}\log x$, with $d_0=1$, hasFor $N=m\beta$ the preceding recurrence relation gives $d_{(m-1)\beta }=b^{m-1}/[{\beta }^{2m-2}((m-1)!{)}^2]$, soThis proves that the logarithm cannot disappear for a special nonzero value of $b$. For $a=0$, reduction of order instead produces the repeated-root Logarithmic Frobenius solution directly.

For a real pair of indicial roots $r_1\ge r_2$ at a regular singular point of a second-order linear differential equation, the second local solution is classified by the gap $\Delta$ and by the recurrence relation for the smaller indicial root. Case I has $\Delta \notin \mathbb{Z}$ and two pure Frobenius solutions. Case II has $\Delta =0$ and an independent Logarithmic Frobenius solution. With a positive integer gap, Case IIIa means the exceptional coefficient equation is inconsistent, so a nonzero multiple of the first Frobenius solution times a logarithm is necessary. Case IIIb means that equation is consistent and leaves a free coefficient, so there are two pure Frobenius solutions. This IIIa/IIIb convention is the one used in the Oxford course notes.^[ref]

The free coefficient in Case IIIb adds a multiple of the first Frobenius solution, as explained by the undetermined coefficient at Frobenius resonance. Stating whether the logarithm occurs is useful alongside the case name because other courses may choose different labels.

When two indicial roots differ by a positive integer, the recurrence for the smaller-root Frobenius solution can have a zero coefficient at the index of the larger root. If the corresponding right-hand side also vanishes, that coefficient is free and adds the larger-root solution; if it does not vanish, a pure Frobenius series is obstructed and a logarithmic term may be required. For $x^2{y}^{\prime \prime }-2x{y}^{\prime }-x^{2}y=0$, the recurrence $n(n-3)a_n=a_{n-2}$ has $a_1=0$ and leaves $a_3$ free. Choosing $a_3=0$ selects the even solution, not the only solution with exponent zero.

With $s=\sqrt{z}$ and $Y(s)=y(s^2)$, the displayed linear differential equation becomes $Y_{ss}+Y=0$ away from zero. Its Frobenius solutions have exponents $0$ and $1/2$: $\cos \sqrt{z}={\sum }_{n\ge 0}(-1{)}^n{z}^n/(2n)!$ and $\sin \sqrt{z}=z^{1/2}{\sum }_{n\ge 0}(-1{)}^n{z}^n/(2n+1)!$. The first is an entire function of $z$; the second needs a local square-root branch. This illustrates how a regular singular point can arise from a degenerate change of independent variable.

A local series solution at a regular singular point, with exponent $r$ satisfying the indicial equation. Resonant exponents can require a logarithmic companion solution.

The equation $x^2{y}^{\prime \prime }-2x{y}^{\prime }+(2-x^2)y=0$ has a regular singular point at zero. Factoring $y=xu$ cancels the Euler derivative terms and leaves $u^{\prime \prime }-u=0$ away from zero. Its analytic solutions $x\cosh x$ and $x\sinh x$ extend across zero. Their coefficient recurrence $(n-1)(n-2)c_n=c_{n-2}$ has independent free coefficients $c_1,c_2$, giving two entire solutions without a logarithmic branch despite integer-separated indicial roots.

A terminating Frobenius series is a Frobenius method solution whose analytic power series factor has only finitely many nonzero coefficients. Its coefficient linear recurrence relation must allow the first omitted coefficient and all following coefficients to be zero. At an integer separation of characteristic exponents at a regular singular point, a resonant coefficient equation can become $0=0$, permitting such a choice rather than forcing a logarithmic term. With exponent zero, termination produces a polynomial solution.

For $y^{\prime \prime }+P(z)y^{\prime }+Q(z)y=0$, the finite point $z_0$ is ordinary when $P$ and $Q$ are holomorphic at $z_0$.

Substitution of a Frobenius behavior $(z-z_0{)}^{\rho }$ into the leading singular terms gives the indicial equation and its characteristic exponents.

For$z_0$ is a regular singular point when $(z-z_0)P(z)$ and $(z-z_0{)}^{2}Q(z)$ are analytic at $z_0$.

Under $\zeta =1/z$, the point $z=\infty$ is regular singular forexactly when $zP(z)$ and $z^{2}Q(z)$ are holomorphic functions of $1/z$ near $1/z=0$.

When a second-order equation has a repeated indicial root and only one independent Frobenius series, a second solution has the local form

A singular point of a linear differential equation is irregular when it fails the regular-singular criterion. For $y^{\prime \prime }+P(x)y^{\prime }+Q(x)y=0$, a pole of order greater than one in $P$ or greater than two in $Q$ makes the point irregular.

For a first-order meromorphic linear ordinary differential equation $Y_t=A(t)Y$, with a coefficient pole of order $p>1$ in a fixed local coordinate and gauge, its Poincaré rank is $p-1$. At infinity one must transform the differential system, not just the matrix-valued function: $Y_{\lambda }=A_0{\lambda }^{m}Y+\cdots$ becomes $Y_t=-A_0{t}^{-m-2}Y+\cdots$ under $t=1/\lambda$, giving rank $m+1$. A singular gauge can lower an apparent rank; distinct leading eigenvalues prevent the elementary degeneracy in the unramified setting.

Riemann's differential equation is a second-order ordinary differential equation with three regular singular points, normalized to zero, one and infinity by a Möbius transformation. Its local characteristic exponents are encoded by the Papperitz symbol and constrained by the Fuchs relation.

A Papperitz or P-symbol lists the three regular singular points of a second-order equation and the two characteristic exponents at each.

This symbol records the local exponents of Riemann's differential equation.

For a second-order equation with three regular singular points, the sum of all six characteristic exponents is one.

For an order-$n$ Fuchsian differential equation with $r$ finite singular points and infinity, the sum of all characteristic exponents is $n(n-1)(r-1)/2$. The three-singularity second-order case has $n=2$ and $r=2$.

A Möbius change of independent variable permutes the singular points while carrying their characteristic exponent pairs with them.

Multiplying a solution by $(z-z_0{)}^{\lambda }$ adds $\lambda$ to both exponents at the finite point $z_0$ and subtracts $\lambda$ from both exponents at infinity, under the convention that an exponent $\rho$ at infinity corresponds to behavior $z^{-\rho }$.

The Gauss equation has singularities at $0,1,\infty$ and normalized exponent-zero solution $F(A,B;C;u)$ at the origin.

For the Gauss hypergeometric equation with parameters $\alpha ,\beta ,\gamma$, a Frobenius solution $y=x^{\sigma }{\sum }_{n\ge 0}{a}_n{x}^n$, $a_0\ne 0$, has indicial roots $\sigma =0$ and $\sigma =1-\gamma$. Matching each successive power gives the displayed undivided recurrence. The usual ratio formula is valid only when its left coefficient is nonzero. At integer-separated roots, a zero coefficient requires a compatibility check as in an undetermined coefficient at Frobenius resonance; otherwise a logarithmic companion may be necessary. For example, $\gamma =0,\sigma =0$ gives $0=\alpha \beta a_0$ at $n=1$, obstructing that exponent-zero series when $\alpha \beta \ne 0$.

Multiply a solution with parameters $(c-a,c-b;c)$ by $(1-z{)}^{c-a-b}$. Its local exponents become exactly those of the Gauss hypergeometric equation with parameters $(a,b;c)$: those at one shift by $c-a-b$, and those at infinity shift by its negative. Direct differentiation gives the same differential equation. Near zero, choose the factor with value one; uniqueness of the analytic normalized solution, for $c$ not a nonpositive integer, gives the identity. Compatible analytic continuation extends it to other domains and parameters where the normalized functions are defined.

Put $d=c-a-b\notin \mathbb{Z}$ and choose a branch of $(1-z{)}^d$. The solution regular at zero isThe two characteristic exponents at a regular singular point are $0,d$. The Euler integral for the hypergeometric function determines the first coefficient by a finite endpoint limit and the second by scaling the singular endpoint; analytic continuation extends the parameter ranges.

The Gauss hypergeometric function is the normalized analytic solution at zero,initially for $|z|<1$ and $c\notin \{0,-1,-2,\dots \}$, followed by analytic continuation where possible.

The ordinary hypergeometric function is ${}_2{F}_1(a,b;c;z)={\sum }_{n\ge 0}(a{)}_n(b{)}_n{z}^n/((c{)}_{n}n!)$ near zero for admissible $c$. It solves the Gauss hypergeometric equation and extends by analytic continuation.

For $\Re c>\Re b>0$ and a fixed branch away from the cut,Expand $(1-zs{)}^{-a}$ for $|z|<1$ and integrate termwise using the beta function to recover the Gauss hypergeometric function series. Analytic continuation extends the identity in $z$.

With compatible branches and initially near $z=0$,It follows by applying a Möbius transformation and a dependent-variable rescaling to the same Papperitz symbol, then matching the normalized analytic solution at zero.

When $C$ is not an integer, a second local solution at zero is $u^{1-C}F(A-C+1,B-C+1;2-C;u)$.

Away from resonant parameter cases, two independent local solutions at infinity areand the expression obtained by exchanging $a$ and $b$. Every analytic continuation of a hypergeometric solution into their common domain is a constant linear combination of this basis.

The binomial series gives $F(A,C;C;u)=(1-u{)}^{-A}$ wherever the defining series converges, followed by analytic continuation.

Special parameter choices can identify a hypergeometric solution with an elementary solution of the same second-order differential equation by matching its local normalization.

For compatible local branches near $x=0$,It follows by transforming the harmonic oscillator equation $y^{\prime \prime }+k^{2}y=0$ into the Gauss hypergeometric equation and selecting the solution with value one and derivative zero at the origin.

For $k\ne 0$ and compatible local branches near $x=0$,This is obtained from the second local hypergeometric solution at zero and its normalization.