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This method enables one to compute a [[Fundamental system of solutions|fundamental system of solutions]] for a holomorphic differential equation near a regular singular point (cf. also [[Singular point|Singular point]]).
 
This method enables one to compute a [[Fundamental system of solutions|fundamental system of solutions]] for a holomorphic differential equation near a regular singular point (cf. also [[Singular point|Singular point]]).
  
 
Suppose one is given a [[Linear differential operator|linear differential operator]]
 
Suppose one is given a [[Linear differential operator|linear differential operator]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} L = \sum _ { n = 0 } ^ { N } a ^ { [ n ] } ( z ) z ^ { n } \left( \frac { d } { d z } \right) ^ { n }, \end{equation}
  
where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202102.png" /> and some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202103.png" />, the functions
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where for $n = 0 , \ldots , N$ and some $r &gt; 0$, the functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202104.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} a ^ { [ n ] } ( z ) = \sum _ { i = 0 } ^ { \infty } a _ { i } ^ { n } z ^ { i } \end{equation}
  
are holomorphic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202106.png" /> (cf. also [[Analytic function|Analytic function]]). The point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202107.png" /> is called a regular singular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202108.png" />. Formula (a1) gives the differential operator in its Frobenius normal form if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f1202109.png" />.
+
are holomorphic for $| z | &lt; r$ and $a ^ { N_ 0} \neq 0$ (cf. also [[Analytic function|Analytic function]]). The point $z = 0$ is called a regular singular point of $L$. Formula (a1) gives the differential operator in its Frobenius normal form if $a ^ { [ N ] } ( z ) \equiv 1$.
  
 
The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation
 
The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} L ( u ) = 0 \end{equation}
  
in the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021011.png" /> near the regular singular point at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021012.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021013.png" />, and for an equation in normal form, actually <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021014.png" />. The cut along some ray is introduced because the solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021015.png" /> are expected to have an essential singularity at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021016.png" />.
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in the domain $\{ z \in \mathbf{C} : | z | &lt; \epsilon \} \backslash ( - \infty , 0 ]$ near the regular singular point at $z = 0$. Here, $\epsilon &gt; 0$, and for an equation in normal form, actually $\epsilon \geq r$. The cut along some ray is introduced because the solutions $u$ are expected to have an essential singularity at $z = 0$.
  
 
The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation
 
The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \tag{a4} L _ { 0 } ( u ) = 0, \end{equation}
  
where the differential operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021018.png" /> is made from (a1) by retaining only the leading terms. The Euler–Cauchy equation can be solved by taking the guess <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021019.png" /> with unknown parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021020.png" />. One gets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021021.png" /> with the indicial polynomial
+
where the differential operator $L_0$ is made from (a1) by retaining only the leading terms. The Euler–Cauchy equation can be solved by taking the guess $z = u ^ { \lambda }$ with unknown parameter $\lambda \in \mathbf{C}$. One gets $L _ { 0 } ( u ^ { \lambda } ) = \pi ( \lambda ) z ^ { \lambda }$ with the indicial polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a5} \pi ( \lambda ) = \sum _ { n = 0 } ^ { N } ( \lambda + n ) ( \lambda + n - 1 ) \ldots ( \lambda + 1 ) a ^ { n _0} = \end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021023.png" /></td> </tr></table>
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\begin{equation*} = a _ { 0 } ^ { N } \prod _ { i = 1 } ^ { \nu } ( \lambda - \lambda _ { i } ) ^ { n _ { i } }. \end{equation*}
  
In the following, the zeros <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021024.png" /> of the indicial polynomial will be ordered by requiring
+
In the following, the zeros $\lambda _ { i }$ of the indicial polynomial will be ordered by requiring
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021025.png" /></td> </tr></table>
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\begin{equation*} \operatorname { Re } \lambda _ { 1 } \geq \ldots \geq \operatorname { Re } \lambda _ { \nu }. \end{equation*}
  
It is assumed that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021026.png" /> roots are different and one denotes their multiplicities by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021027.png" />.
+
It is assumed that all $\nu$ roots are different and one denotes their multiplicities by $n_i$.
  
 
The method of Frobenius starts with the guess
 
The method of Frobenius starts with the guess
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \tag{a6} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation}
  
with an undetermined parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021029.png" />. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021030.png" /> have to be calculated by requiring that
+
with an undetermined parameter $\lambda \in \mathbf{C}$. The coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021030.png"/> have to be calculated by requiring that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
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\begin{equation} \tag{a7} L ( u ( z , \lambda ) ) = \pi ( \lambda ) z ^ { \lambda }. \end{equation}
  
This requirement leads to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021032.png" /> and
+
This requirement leads to $c _ { 0 } \equiv 1$ and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
+
\begin{equation} \tag{a8} c _ { j } ( \lambda ) = - \sum _ { k = 0 } ^ { j - 1 } \frac { c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) } { \pi ( \lambda + j ) } \end{equation}
  
as a recursion formula for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021035.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021036.png" /> are polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021037.png" /> of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021038.png" />, which are given below.
+
as a recursion formula for $c_{j}$ for all $j \geq 1$. Here, $p _ { j } ( \lambda )$ are polynomials in $\lambda$ of degree at most $N$, which are given below.
  
The easy generic case occurs if the indicial polynomial has only simple zeros and their differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021039.png" /> are never integer valued. Under these assumptions, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021040.png" /> functions
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The easy generic case occurs if the indicial polynomial has only simple zeros and their differences $\lambda _ { i } - \lambda _ { j }$ are never integer valued. Under these assumptions, the $N$ functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021041.png" /></td> </tr></table>
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\begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots , \ldots , u ( z , \lambda _ { N } ) = z ^ { \lambda _ { N } } +\dots \end{equation*}
  
 
are a fundamental system of solutions of (a3).
 
are a fundamental system of solutions of (a3).
  
 
==Complications.==
 
==Complications.==
Complications can arise if the generic assumption made above is not satisfied. Putting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021042.png" /> in (a6), obtaining solutions of (a3) can be impossible because of poles of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021043.png" />. These solutions are rational functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021044.png" /> with possible poles at the poles of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021045.png" /> as well as at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021046.png" />.
+
Complications can arise if the generic assumption made above is not satisfied. Putting $\lambda = \lambda _ { i }$ in (a6), obtaining solutions of (a3) can be impossible because of poles of the coefficients $c_j ( \lambda )$. These solutions are rational functions of $\lambda$ with possible poles at the poles of $c _ { 1 } ( \lambda ) , \ldots , c _ { j  - 1} ( \lambda )$ as well as at $\lambda _ { 1 } + j , \ldots , \lambda _ { \nu } + j$.
  
The poles are compensated for by multiplying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021047.png" /> at first with powers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021048.png" /> and differentiation by the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021049.png" /> before setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021050.png" />.
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The poles are compensated for by multiplying $u ( z , \lambda )$ at first with powers of $\lambda - \lambda _ { i }$ and differentiation by the parameter $\lambda$ before setting $\lambda = \lambda _ { i }$.
  
Since the general situation is rather complex, two special cases are given first. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021051.png" /> denote the set of natural numbers starting at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021052.png" /> (i.e., excluding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021053.png" />). Note that neither of the special cases below does exclude the simple generic case above.
+
Since the general situation is rather complex, two special cases are given first. Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). Note that neither of the special cases below does exclude the simple generic case above.
  
 
All solutions have expansions of the form
 
All solutions have expansions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021054.png" /></td> </tr></table>
+
\begin{equation*} u _ { i l } = z ^ { \lambda _ { i } } \sum _ { j = 0 } ^ { l } \sum _ { k = 0 } ^ { \infty } b _ { j k } ( \operatorname { log } z ) ^ { j } z ^ { k }. \end{equation*}
  
The leading term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021055.png" /> is useful as a marker for the different solutions. Because for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021057.png" />, all leading terms are different, the method of Frobenius does indeed yield a fundamental system of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021058.png" /> linearly independent solutions of the differential equation (a3).
+
The leading term $b _ { l0 } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { i } }$ is useful as a marker for the different solutions. Because for $i = 1 , \dots , \nu$ and $l = 0 , \dots , n _ { i } - 1$, all leading terms are different, the method of Frobenius does indeed yield a fundamental system of $N$ linearly independent solutions of the differential equation (a3).
  
 
===Special case 1.===
 
===Special case 1.===
For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021059.png" />, the zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021060.png" /> of the indicial polynomial has multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021061.png" />, but none of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021062.png" /> is a natural number.
+
For any $i = 1 , \dots , \nu$, the zero $\lambda _ { i }$ of the indicial polynomial has multiplicity $n _ { i } \geq 1$, but none of the numbers $\lambda _ { 1 } - \lambda _ { i } , \ldots , \lambda _ { i - 1 } - \lambda _ { i }$ is a natural number.
  
 
In this case, the functions
 
In this case, the functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021063.png" /></td> </tr></table>
+
\begin{equation*} u ( z , \lambda _ { i } ) = z ^ { \lambda _ { i } } + \ldots , \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021064.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial } { \partial \lambda } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) z ^ { \lambda_i }  +\dots \dots \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021065.png" /></td> </tr></table>
+
\begin{equation*} \left( \frac { \partial } { \partial \lambda } \right) ^ { ( n _ { i } - 1 ) } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) ^ { n _ { i } - 1 } z ^ { \lambda _ { i } } +\dots \end{equation*}
  
are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021066.png" /> linearly independent solutions of the differential equation (a3).
+
are $n_i$ linearly independent solutions of the differential equation (a3).
  
 
===Special case 2.===
 
===Special case 2.===
Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021067.png" />.
+
Suppose $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$.
  
 
Then the functions
 
Then the functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021068.png" /></td> </tr></table>
+
\begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { n _ { 1 } + l } [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ^ { n _ { 1 } } ] = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021069.png" /></td> </tr></table>
+
\begin{equation*} = \frac { ( n _ { 1 } + l ) ! } { l ! } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { 2 } } + \ldots, \end{equation*}
  
all with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021070.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021071.png" />, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021072.png" /> linearly independent solutions of the differential equation (a3). The solution for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021073.png" /> may contain logarithmic terms in the higher powers, starting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021074.png" />.
+
all with $\lambda = \lambda _ { 2 }$ and $l = 0 , \dots , n _ { 2 } - 1$, are $n_{2}$ linearly independent solutions of the differential equation (a3). The solution for $l = 0$ may contain logarithmic terms in the higher powers, starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$.
  
 
===Special case 3.===
 
===Special case 3.===
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021075.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021076.png" /> be a zero of the indicial polynomial of multiplicity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021077.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021078.png" />.
+
Let $1 \leq j \leq \nu$ and let $\lambda _ { i }$ be a zero of the indicial polynomial of multiplicity $n_i$ for $i = 1 , \dots , j - 1$.
  
In this case, define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021079.png" /> to be the sum of those multiplicities for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021080.png" />. Hence,
+
In this case, define $m_j$ to be the sum of those multiplicities for which $\lambda _ { i } - \lambda _ { j } \in \mathbf{N}$. Hence,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021081.png" /></td> </tr></table>
+
\begin{equation*} m _ { j } = \sum \{ n _ { i } : 1 \leq i &lt; j \  \text{ and } \ \lambda _ { i } - \lambda _ { j } \in \mathbf{N} \}. \end{equation*}
  
 
The functions
 
The functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021082.png" /></td> </tr></table>
+
\begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { m _ { j } + l } \left[ u ( z , \lambda ) ( \lambda - \lambda _ { j } ) ^ { m _ { j } } \right] = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021083.png" /></td> </tr></table>
+
\begin{equation*} = \frac { ( m _ { j } + l ) ! } { l ! } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { j } } + \ldots, \end{equation*}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021084.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021085.png" />, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021086.png" /> linearly independent solutions of the differential equation (a3).
+
with $l = 0 , \dots , n _ { j } - 1$ and $\lambda = \lambda _ { j }$, are $n_j$ linearly independent solutions of the differential equation (a3).
  
 
The method looks simpler in the most common case of a differential operator
 
The method looks simpler in the most common case of a differential operator
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021087.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
+
\begin{equation} \tag{a9} L = a ^ { [ 2 ] } ( z ) z ^ { 2 } \left( \frac { d } { d z } \right) ^ { 2 } + a ^ { [ 1 ] } ( z ) z \left( \frac { d } { d z } \right) + a ^ { [ 0 ] } ( z ). \end{equation}
  
Here, one has to to assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021088.png" /> to obtain a regular singular point. The indicial polynomial is simply
+
Here, one has to to assume that $a ^ { 2_0 } \neq 0$ to obtain a regular singular point. The indicial polynomial is simply
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021089.png" /></td> </tr></table>
+
\begin{equation*} \pi ( \lambda ) = ( \lambda + 2 ) ( \lambda + 1 ) a ^ { 2_0 } + ( \lambda + 1 ) a ^ { 1_0 }  + a ^ { 0_0 } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021090.png" /></td> </tr></table>
+
\begin{equation*} = a ^ { 2 } o ( \lambda - \lambda _ { 1 } ) ( \lambda - \lambda _ { 2 } ). \end{equation*}
  
 
Only two special cases can occur:
 
Only two special cases can occur:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021091.png" />. The functions
+
1) $\lambda _ { 1 } = \lambda _ { 2 }$. The functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021092.png" /></td> </tr></table>
+
\begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021093.png" /></td> </tr></table>
+
\begin{equation*} \frac { \partial u } { \partial \lambda } ( z , \lambda _ { 1 } ) = ( \operatorname { log } z ) z ^ { \lambda _ { 1 } } \end{equation*}
  
 
are a fundamental system.
 
are a fundamental system.
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021094.png" />. The functions
+
2) $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. The functions
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021095.png" /></td> </tr></table>
+
\begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots, \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021096.png" /></td> </tr></table>
+
\begin{equation*} ( \frac { \partial } { \partial \lambda } ) [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ] = z ^ { \lambda_2 }  + \ldots , \end{equation*}
  
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021097.png" /> in the second function, are two linearly independent solutions of the differential equation (a9). The second solution can contain logarithmic terms in the higher powers starting with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021098.png" />.
+
with $\lambda = \lambda _ { 2 }$ in the second function, are two linearly independent solutions of the differential equation (a9). The second solution can contain logarithmic terms in the higher powers starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$.
  
 
The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. also [[Fuchsian equation|Fuchsian equation]]). A similar method of solution can be used for matrix equations of the first order, too. An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers.
 
The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. also [[Fuchsian equation|Fuchsian equation]]). A similar method of solution can be used for matrix equations of the first order, too. An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers.
  
==Computation of the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f12021099.png" />.==
+
==Computation of the polynomials $p _ { j } ( \lambda )$.==
 
In the guess
 
In the guess
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210100.png" /></td> </tr></table>
+
\begin{equation*} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation*}
  
the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210101.png" /> have to be calculated from the requirement (a7). Indeed (a1) and (a2) imply
+
the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210101.png"/> have to be calculated from the requirement (a7). Indeed (a1) and (a2) imply
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210102.png" /></td> </tr></table>
+
\begin{equation*} L ( u ( z , \lambda ) ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210103.png" /></td> </tr></table>
+
\begin{equation*} = [ \sum _ { i = 0 } ^ { \infty } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n + i } ( \frac { \partial } { \partial z } ) ^ { n } ] [ \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { \lambda + k } ] = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210104.png" /></td> </tr></table>
+
\begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n } \left( \frac { \partial } { \partial z } \right) ^ { n } z ^ { \lambda + k } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210105.png" /></td> </tr></table>
+
\begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } p _ { i } ( \lambda + k ) z ^ { \lambda + k } = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210106.png" /></td> </tr></table>
+
\begin{equation*} = z ^ { \lambda } \sum _ { j = 0 } ^ { \infty } z ^ { j } \left[ \sum _ { i + k = j } c _ { k } ( \lambda ) p _ { i } ( \lambda + k ) \right] = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210107.png" /></td> </tr></table>
+
\begin{equation*} = c _ { 0 } z ^ { \lambda } \pi ( \lambda ) + \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210108.png" /></td> </tr></table>
+
\begin{equation*} + z ^ { \lambda } \sum _ { j = 1 } ^ { \infty } z ^ { j } \left[ c _ { j } ( \lambda ) \pi ( \lambda + j ) + \sum _ { k = 0 } ^ { j - 1 } c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) \right]. \end{equation*}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210109.png" /> are polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210110.png" /> determined by setting
+
Here, $p _ { i } ( \lambda )$ are polynomials of degree at most $N$ determined by setting
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210111.png" /></td> </tr></table>
+
\begin{equation*} p _ { i } ( z ) z ^ { \lambda } = \sum _ { n = 0 } ^ { N } a ^ { n _ { i } } z ^ { n } ( \frac { \partial } { \partial z } ) ^ { n } z ^ { \lambda }. \end{equation*}
  
Because of (a7), one finds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120210/f120210112.png" /> and the recursion formula (a8).
+
Because of (a7), one finds $c _ { 0 } \equiv 1$ and the recursion formula (a8).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Redheffer,  "Differential equations, theory and applications" , Jones and Bartlett  (1991)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Rothe,  "A variant of Frobenius' method for the Emden–Fowler equation"  ''Applicable Anal.'' , '''66'''  (1997)  pp. 217–245</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Zwillinger,  "Handbook of differential equations" , Acad. Press  (1989)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  R. Redheffer,  "Differential equations, theory and applications" , Jones and Bartlett  (1991)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  F. Rothe,  "A variant of Frobenius' method for the Emden–Fowler equation"  ''Applicable Anal.'' , '''66'''  (1997)  pp. 217–245</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  D. Zwillinger,  "Handbook of differential equations" , Acad. Press  (1989)</td></tr></table>

Revision as of 16:58, 1 July 2020

This method enables one to compute a fundamental system of solutions for a holomorphic differential equation near a regular singular point (cf. also Singular point).

Suppose one is given a linear differential operator

\begin{equation} \tag{a1} L = \sum _ { n = 0 } ^ { N } a ^ { [ n ] } ( z ) z ^ { n } \left( \frac { d } { d z } \right) ^ { n }, \end{equation}

where for $n = 0 , \ldots , N$ and some $r > 0$, the functions

\begin{equation} \tag{a2} a ^ { [ n ] } ( z ) = \sum _ { i = 0 } ^ { \infty } a _ { i } ^ { n } z ^ { i } \end{equation}

are holomorphic for $| z | < r$ and $a ^ { N_ 0} \neq 0$ (cf. also Analytic function). The point $z = 0$ is called a regular singular point of $L$. Formula (a1) gives the differential operator in its Frobenius normal form if $a ^ { [ N ] } ( z ) \equiv 1$.

The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation

\begin{equation} \tag{a3} L ( u ) = 0 \end{equation}

in the domain $\{ z \in \mathbf{C} : | z | < \epsilon \} \backslash ( - \infty , 0 ]$ near the regular singular point at $z = 0$. Here, $\epsilon > 0$, and for an equation in normal form, actually $\epsilon \geq r$. The cut along some ray is introduced because the solutions $u$ are expected to have an essential singularity at $z = 0$.

The Frobenius method is a generalization of the treatment of the simpler Euler–Cauchy equation

\begin{equation} \tag{a4} L _ { 0 } ( u ) = 0, \end{equation}

where the differential operator $L_0$ is made from (a1) by retaining only the leading terms. The Euler–Cauchy equation can be solved by taking the guess $z = u ^ { \lambda }$ with unknown parameter $\lambda \in \mathbf{C}$. One gets $L _ { 0 } ( u ^ { \lambda } ) = \pi ( \lambda ) z ^ { \lambda }$ with the indicial polynomial

\begin{equation} \tag{a5} \pi ( \lambda ) = \sum _ { n = 0 } ^ { N } ( \lambda + n ) ( \lambda + n - 1 ) \ldots ( \lambda + 1 ) a ^ { n _0} = \end{equation}

\begin{equation*} = a _ { 0 } ^ { N } \prod _ { i = 1 } ^ { \nu } ( \lambda - \lambda _ { i } ) ^ { n _ { i } }. \end{equation*}

In the following, the zeros $\lambda _ { i }$ of the indicial polynomial will be ordered by requiring

\begin{equation*} \operatorname { Re } \lambda _ { 1 } \geq \ldots \geq \operatorname { Re } \lambda _ { \nu }. \end{equation*}

It is assumed that all $\nu$ roots are different and one denotes their multiplicities by $n_i$.

The method of Frobenius starts with the guess

\begin{equation} \tag{a6} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation}

with an undetermined parameter $\lambda \in \mathbf{C}$. The coefficients have to be calculated by requiring that

\begin{equation} \tag{a7} L ( u ( z , \lambda ) ) = \pi ( \lambda ) z ^ { \lambda }. \end{equation}

This requirement leads to $c _ { 0 } \equiv 1$ and

\begin{equation} \tag{a8} c _ { j } ( \lambda ) = - \sum _ { k = 0 } ^ { j - 1 } \frac { c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) } { \pi ( \lambda + j ) } \end{equation}

as a recursion formula for $c_{j}$ for all $j \geq 1$. Here, $p _ { j } ( \lambda )$ are polynomials in $\lambda$ of degree at most $N$, which are given below.

The easy generic case occurs if the indicial polynomial has only simple zeros and their differences $\lambda _ { i } - \lambda _ { j }$ are never integer valued. Under these assumptions, the $N$ functions

\begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots , \ldots , u ( z , \lambda _ { N } ) = z ^ { \lambda _ { N } } +\dots \end{equation*}

are a fundamental system of solutions of (a3).

Complications.

Complications can arise if the generic assumption made above is not satisfied. Putting $\lambda = \lambda _ { i }$ in (a6), obtaining solutions of (a3) can be impossible because of poles of the coefficients $c_j ( \lambda )$. These solutions are rational functions of $\lambda$ with possible poles at the poles of $c _ { 1 } ( \lambda ) , \ldots , c _ { j - 1} ( \lambda )$ as well as at $\lambda _ { 1 } + j , \ldots , \lambda _ { \nu } + j$.

The poles are compensated for by multiplying $u ( z , \lambda )$ at first with powers of $\lambda - \lambda _ { i }$ and differentiation by the parameter $\lambda$ before setting $\lambda = \lambda _ { i }$.

Since the general situation is rather complex, two special cases are given first. Let $\mathbf{N}$ denote the set of natural numbers starting at $1$ (i.e., excluding $0$). Note that neither of the special cases below does exclude the simple generic case above.

All solutions have expansions of the form

\begin{equation*} u _ { i l } = z ^ { \lambda _ { i } } \sum _ { j = 0 } ^ { l } \sum _ { k = 0 } ^ { \infty } b _ { j k } ( \operatorname { log } z ) ^ { j } z ^ { k }. \end{equation*}

The leading term $b _ { l0 } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { i } }$ is useful as a marker for the different solutions. Because for $i = 1 , \dots , \nu$ and $l = 0 , \dots , n _ { i } - 1$, all leading terms are different, the method of Frobenius does indeed yield a fundamental system of $N$ linearly independent solutions of the differential equation (a3).

Special case 1.

For any $i = 1 , \dots , \nu$, the zero $\lambda _ { i }$ of the indicial polynomial has multiplicity $n _ { i } \geq 1$, but none of the numbers $\lambda _ { 1 } - \lambda _ { i } , \ldots , \lambda _ { i - 1 } - \lambda _ { i }$ is a natural number.

In this case, the functions

\begin{equation*} u ( z , \lambda _ { i } ) = z ^ { \lambda _ { i } } + \ldots , \end{equation*}

\begin{equation*} \frac { \partial } { \partial \lambda } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) z ^ { \lambda_i } +\dots \dots \end{equation*}

\begin{equation*} \left( \frac { \partial } { \partial \lambda } \right) ^ { ( n _ { i } - 1 ) } u ( z , \lambda _ { i } ) = ( \operatorname { log } z ) ^ { n _ { i } - 1 } z ^ { \lambda _ { i } } +\dots \end{equation*}

are $n_i$ linearly independent solutions of the differential equation (a3).

Special case 2.

Suppose $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$.

Then the functions

\begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { n _ { 1 } + l } [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ^ { n _ { 1 } } ] = \end{equation*}

\begin{equation*} = \frac { ( n _ { 1 } + l ) ! } { l ! } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { 2 } } + \ldots, \end{equation*}

all with $\lambda = \lambda _ { 2 }$ and $l = 0 , \dots , n _ { 2 } - 1$, are $n_{2}$ linearly independent solutions of the differential equation (a3). The solution for $l = 0$ may contain logarithmic terms in the higher powers, starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$.

Special case 3.

Let $1 \leq j \leq \nu$ and let $\lambda _ { i }$ be a zero of the indicial polynomial of multiplicity $n_i$ for $i = 1 , \dots , j - 1$.

In this case, define $m_j$ to be the sum of those multiplicities for which $\lambda _ { i } - \lambda _ { j } \in \mathbf{N}$. Hence,

\begin{equation*} m _ { j } = \sum \{ n _ { i } : 1 \leq i < j \ \text{ and } \ \lambda _ { i } - \lambda _ { j } \in \mathbf{N} \}. \end{equation*}

The functions

\begin{equation*} ( \frac { \partial } { \partial \lambda } ) ^ { m _ { j } + l } \left[ u ( z , \lambda ) ( \lambda - \lambda _ { j } ) ^ { m _ { j } } \right] = \end{equation*}

\begin{equation*} = \frac { ( m _ { j } + l ) ! } { l ! } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { j } } + \ldots, \end{equation*}

with $l = 0 , \dots , n _ { j } - 1$ and $\lambda = \lambda _ { j }$, are $n_j$ linearly independent solutions of the differential equation (a3).

The method looks simpler in the most common case of a differential operator

\begin{equation} \tag{a9} L = a ^ { [ 2 ] } ( z ) z ^ { 2 } \left( \frac { d } { d z } \right) ^ { 2 } + a ^ { [ 1 ] } ( z ) z \left( \frac { d } { d z } \right) + a ^ { [ 0 ] } ( z ). \end{equation}

Here, one has to to assume that $a ^ { 2_0 } \neq 0$ to obtain a regular singular point. The indicial polynomial is simply

\begin{equation*} \pi ( \lambda ) = ( \lambda + 2 ) ( \lambda + 1 ) a ^ { 2_0 } + ( \lambda + 1 ) a ^ { 1_0 } + a ^ { 0_0 } = \end{equation*}

\begin{equation*} = a ^ { 2 } o ( \lambda - \lambda _ { 1 } ) ( \lambda - \lambda _ { 2 } ). \end{equation*}

Only two special cases can occur:

1) $\lambda _ { 1 } = \lambda _ { 2 }$. The functions

\begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots, \end{equation*}

\begin{equation*} \frac { \partial u } { \partial \lambda } ( z , \lambda _ { 1 } ) = ( \operatorname { log } z ) z ^ { \lambda _ { 1 } } \end{equation*}

are a fundamental system.

2) $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. The functions

\begin{equation*} u ( z , \lambda _ { 1 } ) = z ^ { \lambda _ { 1 } } + \ldots, \end{equation*}

\begin{equation*} ( \frac { \partial } { \partial \lambda } ) [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ] = z ^ { \lambda_2 } + \ldots , \end{equation*}

with $\lambda = \lambda _ { 2 }$ in the second function, are two linearly independent solutions of the differential equation (a9). The second solution can contain logarithmic terms in the higher powers starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$.

The Frobenius method has been used very successfully to develop a theory of analytic differential equations, especially for the equations of Fuchsian type, where all singular points assumed to be regular (cf. also Fuchsian equation). A similar method of solution can be used for matrix equations of the first order, too. An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers.

Computation of the polynomials $p _ { j } ( \lambda )$.

In the guess

\begin{equation*} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k }, \end{equation*}

the coefficients have to be calculated from the requirement (a7). Indeed (a1) and (a2) imply

\begin{equation*} L ( u ( z , \lambda ) ) = \end{equation*}

\begin{equation*} = [ \sum _ { i = 0 } ^ { \infty } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n + i } ( \frac { \partial } { \partial z } ) ^ { n } ] [ \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { \lambda + k } ] = \end{equation*}

\begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } \sum _ { n = 0 } ^ { N } a _ { i } ^ { n } z ^ { n } \left( \frac { \partial } { \partial z } \right) ^ { n } z ^ { \lambda + k } = \end{equation*}

\begin{equation*} = \sum _ { i = 0 } ^ { \infty } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { i } p _ { i } ( \lambda + k ) z ^ { \lambda + k } = \end{equation*}

\begin{equation*} = z ^ { \lambda } \sum _ { j = 0 } ^ { \infty } z ^ { j } \left[ \sum _ { i + k = j } c _ { k } ( \lambda ) p _ { i } ( \lambda + k ) \right] = \end{equation*}

\begin{equation*} = c _ { 0 } z ^ { \lambda } \pi ( \lambda ) + \end{equation*}

\begin{equation*} + z ^ { \lambda } \sum _ { j = 1 } ^ { \infty } z ^ { j } \left[ c _ { j } ( \lambda ) \pi ( \lambda + j ) + \sum _ { k = 0 } ^ { j - 1 } c _ { k } ( \lambda ) p _ { j - k } ( \lambda + k ) \right]. \end{equation*}

Here, $p _ { i } ( \lambda )$ are polynomials of degree at most $N$ determined by setting

\begin{equation*} p _ { i } ( z ) z ^ { \lambda } = \sum _ { n = 0 } ^ { N } a ^ { n _ { i } } z ^ { n } ( \frac { \partial } { \partial z } ) ^ { n } z ^ { \lambda }. \end{equation*}

Because of (a7), one finds $c _ { 0 } \equiv 1$ and the recursion formula (a8).

References

[a1] R. Redheffer, "Differential equations, theory and applications" , Jones and Bartlett (1991)
[a2] F. Rothe, "A variant of Frobenius' method for the Emden–Fowler equation" Applicable Anal. , 66 (1997) pp. 217–245
[a3] D. Zwillinger, "Handbook of differential equations" , Acad. Press (1989)
How to Cite This Entry:
Frobenius method. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frobenius_method&oldid=50257
This article was adapted from an original article by Franz Rothe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article