666.7 666.7 666.7 666.7 611.1 611.1 444.4 444.4 444.4 444.4 500 500 388.9 388.9 277.8 as a recursion formula for $c_{j}$ for all $j \geq 1$. /Subtype/Type1 /LastChar 196 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 /FirstChar 33 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 875 531.3 531.3 875 849.5 799.8 812.5 862.3 738.4 707.2 884.3 879.6 419 581 880.8 << 589.1 483.8 427.7 555.4 505 556.5 425.2 527.8 579.5 613.4 636.6 272] Method of Frobenius: Equal Roots to the Indicial Equation We solve the equation x2y''+3 xy'+H1-xL y=0 using a power series centered at the regular singular point x=0. >> Here, $\epsilon > 0$, and for an equation in normal form, actually $\epsilon \geq r$. The approach does produce special separatrix-type solutions for the Emden–Fowler equation, where the non-linear term contains only powers. 38 0 obj with $\lambda = \lambda _ { 2 }$ in the second function, are two linearly independent solutions of the differential equation (a9). /LastChar 196 endobj Since a change x-x 0 ↦ x of variable brings to the case that the singular point is the origin, we may suppose such a starting situation. /Subtype/Type1 Introduction The “na¨ıve” Frobenius method The general Frobenius method Remarks Under the hypotheses of the theorem, we say that a = 0 is a regular singular point of the ODE. The coefficients have to be calculated by requiring that, \tag{a7} L ( u ( z , \lambda ) ) = \pi ( \lambda ) z ^ { \lambda }. << << 491.3 383.7 615.2 517.4 762.5 598.1 525.2 494.2 349.5 400.2 673.4 531.3 295.1 0 0 This method enables one to compute a fundamental system of solutions for a holomorphic differential equation near a regular singular point (cf. This could happen if r 1 = r 2, or if r 1 = r 2 + N. In the latter case there might, or might not, be two Frobenius solutions. 334 405.1 509.3 291.7 856.5 584.5 470.7 491.4 434.1 441.3 461.2 353.6 557.3 473.4 If r1¡r2= 0, the solution basis of the ODE(1)is given by y1(x) =xr1. stream /BaseFont/LQKHRU+CMSY8 For instance, with r= are holomorphic for $| z | < r$ and $a ^ { N_ 0} \neq 0$ (cf. endobj /FirstChar 33 384.3 611.1 675.9 351.8 384.3 643.5 351.8 1000 675.9 611.1 675.9 643.5 481.5 488 Notice that this last solution is always singular at t = 0, whatever the value of γ1! n: 2. 2n 2, so Frobenius’ method fails. /Type/Font Example 3: x = 0 is an irregular point of the ﬂrst order equation Ly = x2y0 +y = 0 The solution of this ﬂrst order linear equation can be obtained by means of … 27 0 obj The easy generic case occurs if the indicial polynomial has only simple zeros and their differences $\lambda _ { i } - \lambda _ { j }$ are never integer valued. 21 0 obj 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 $\lambda = \lambda _ { i }$. /LastChar 196 /Name/F4 (You should check that zero is really a regular singular point.) n≥2. 1111.1 1511.1 1111.1 1511.1 1111.1 1511.1 1055.6 944.4 472.2 833.3 833.3 833.3 833.3 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 351.8 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 611.1 351.8 351.8 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 Suppose $\lambda _ { 1 } - \lambda _ { 2 } \in \mathbf{N}$. /Type/Font Regular and Irregular Singularities As seen in the preceding example, there are situations in which it is not possible to use Frobenius’ method to obtain a series solution. For the case r= 1, we have a n = a n 1 5n+ 6 = ( 1)na 0 Yn k=1 (5j+ 1) 1; n= 1;2;:::; (36) and for r= 1 5, we have a n = a n 1 5n = ( 1)n 5nn! /Subtype/Type1 Hence, \begin{equation*} m _ { j } = \sum \{ n _ { i } : 1 \leq i < j \ \text{ and } \ \lambda _ { i } - \lambda _ { j } \in \mathbf{N} \}. The Set-Up The Calculations and Examples The Main Theorems Inserting the Series into the DE Getting the Coe cients Observations Coe cients We have, rst of all, F (r )=r (r 1 )+p 0 r +q 0 =0 ; the indicial equation. << One gets $L _ { 0 } ( u ^ { \lambda } ) = \pi ( \lambda ) z ^ { \lambda }$ with the indicial polynomial, \tag{a5} \pi ( \lambda ) = \sum _ { n = 0 } ^ { N } ( \lambda + n ) ( \lambda + n - 1 ) \ldots ( \lambda + 1 ) a ^ { n _0} = , \begin{equation*} = a _ { 0 } ^ { N } \prod _ { i = 1 } ^ { \nu } ( \lambda - \lambda _ { i } ) ^ { n _ { i } }. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 663.6 885.4 826.4 736.8 endobj EnMath B, ESE 319-01, Spring 2015 Lecture 4: Frobenius Step-by-Step Jan. 23, 2015 I expect you to >> There is a theorem dealing /Type/Font 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 The leading term $b _ { l0 } ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { i } }$ is useful as a marker for the different solutions. You were also shown how to integrate the equation to … Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory.He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. /Name/F6 /FirstChar 33 also Fuchsian equation). The cut along some ray is introduced because the solutions $u$ are expected to have an essential singularity at $z = 0$. /Widths[295.1 531.3 885.4 531.3 885.4 826.4 295.1 413.2 413.2 531.3 826.4 295.1 354.2 >> When the roots of initial When the roots of initial equation are real, there is a Frobeni us solution for the larger of the tw o roots. /Length 1951 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 458.3 458.3 416.7 416.7 756.4 705.8 763.6 708.3 708.3 708.3 708.3 708.3 649.3 649.3 472.2 472.2 472.2 472.2 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /Type/Font /BaseFont/KNRCDC+CMMI12 Section 1.1 Frobenius Method In this section, we consider a method to find a general solution to a second order ODE about a singular point, written in either of the two equivalent forms below: $$x^2 y'' + xb(x)y' + c(x) y = 0\label{frobenius-standard-form1}\tag{1.1.1}$$ /Type/Font \end{equation*}. in the domain $\{ z \in \mathbf{C} : | z | < \epsilon \} \backslash ( - \infty , 0 ]$ near the regular singular point at $z = 0$. ���ů�f4[rI�[��l�rC\�7 ����Kn���&��͇�u����#V�Z*NT�&�����m�º��Wx�9�������U]�Z��l�۲.��u���7(���"Z�^d�MwK=�!2��jQ&3I�pݔ��HXE�͖��. This is the extensive document regarding the Frobenius Method. 413.2 590.3 560.8 767.4 560.8 560.8 472.2 531.3 1062.5 531.3 531.3 531.3 0 0 0 0 1000 1000 1055.6 1055.6 1055.6 777.8 666.7 666.7 450 450 450 450 777.8 777.8 0 0 The method of Frobenius gives a series solution of the form y(x) = X∞ n=0 an (x −c)n+s where p or q are singular at x = c. Method does not always give the general solution, the ν = 0 case of Bessel’s equation is an example where it doesn’t. 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 >> /Name/F2 1. 30 0 obj 12 0 obj In the Frobenius method one examines whether the equation (2) allows a series solution of the form. /BaseFont/TBNXTN+CMTI12 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 This article was adapted from an original article by Franz Rothe (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. 597.2 736.1 736.1 527.8 527.8 583.3 583.3 583.3 583.3 750 750 750 750 1044.4 1044.4 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 472.2 472.2 472.2 472.2 583.3 583.3 0 0 472.2 472.2 333.3 555.6 577.8 577.8 597.2 /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 /FontDescriptor 32 0 R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 826.4 295.1 826.4 531.3 826.4 { l ! } 1444.4 555.6 1000 1444.4 472.2 472.2 527.8 527.8 527.8 527.8 666.7 666.7 1000 1000 /Widths[351.8 611.1 1000 611.1 1000 935.2 351.8 481.5 481.5 611.1 935.2 351.8 416.7 947.3 784.1 748.3 631.1 775.5 745.3 602.2 573.9 665 570.8 924.4 812.6 568.1 670.2 783.4 872.8 823.4 619.8 708.3 654.8 0 0 816.7 682.4 596.2 547.3 470.1 429.5 467 533.2 Question: Exercise 3. 489.6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 611.8 816 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 << Computation of the polynomials $p _ { j } (\lambda)$. /FontDescriptor 23 0 R /FirstChar 33 Method of Frobenius Example First Solution Second Solution (Fails) What is the Method of Frobenius? 1. 9 0 obj 324.7 531.3 590.3 295.1 324.7 560.8 295.1 885.4 590.3 531.3 590.3 560.8 414.1 419.1 Application of Frobenius’ method In order to solve (3.5), (3.6) we start from a plausible representation of B x,B y that is In fact Frobenius method is just an extension from the power series method that you add an additional power that may not be an integer to each term in a power series or even add the log term for the assumptions of the solution form of the linear ODEs so that you can find all groups of the linearly independent solutions that in cases of cannot find all groups of the linearly independent solutions … 2≥ − − =−for n n n a an n. Since we begin our evaluation of anat n= 2, this final recursion relation will yield valid values for an(since the denominator is never zero for .) 593.8 500 562.5 1125 562.5 562.5 562.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 >> >> The Frobenius method is useful for calculating a fundamental system for the homogeneous linear differential equation, $$\tag{a3} L ( u ) = 0$$. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 606.7 816 748.3 679.6 728.7 811.3 765.8 571.2 656.3 625 625 937.5 937.5 312.5 343.8 562.5 562.5 562.5 562.5 562.5 849.5 500 574.1 a 0; n= 1;2;:::: (37) In the latter case, the solution y(x) has a closed form expression y(x) = x 15 X1 n=0 ( 1)n 5nn! /FontDescriptor 8 0 R 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 Consider roots r1;r2of the indicial equation(3). /Name/F9 An infinite series of the form in (9) is called a Frobenius series. Commonly, the expansion point can be taken as, resulting in the Maclaurin series (1) Section 8.4 The Frobenius Method 467 where the coefﬁcients a n are determined as in Case (a), and the coefﬁcients α n are found by substituting y(x) = y 2(x) into the differential equation. a 0x endobj This case is an example of a CASE III equation where the method of Frobenius will yield both solutions to the differential equation. /BaseFont/SHKLKE+CMEX10 /Subtype/Type1 450 500 300 300 450 250 800 550 500 500 450 412.5 400 325 525 450 650 450 475 400 << %PDF-1.2 \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 }. ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { 2 } } + \ldots, \end{equation*}. /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 The Method of Frobenius (4.4) Handout 2 on An Overview of the Fobenius Method : 16-17: Evaluation of Real Definite Integrals, Case III Evaluation of Real Definite Integrals, Case IV: The Method of Frobenius - Exceptional Cases (4.4, 4.5, 4.6) 18-19: Theorems for Contour Integration Series and … 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. 300 325 500 500 500 500 500 814.8 450 525 700 700 500 863.4 963.4 750 250 500] /Widths[300 500 800 755.2 800 750 300 400 400 500 750 300 350 300 500 500 500 500 Suppose one is given a linear differential operator, $$\tag{a1} L = \sum _ { n = 0 } ^ { N } a ^ { [ n ] } ( z ) z ^ { n } \left( \frac { d } { d z } \right) ^ { n },$$, where for $n = 0 , \ldots , N$ and some $r > 0$, the functions, $$\tag{a2} a ^ { [ n ] } ( z ) = \sum _ { i = 0 } ^ { \infty } a _ { i } ^ { n } z ^ { i }$$. Since the general situation is rather complex, two special cases are given first. /BaseFont/NPKUUX+CMMI8 However, the method of Frobenius can be extended to the case where , , and are functions that can be represented by power series in on some interval that contains zero, and . A similar method of solution can be used for matrix equations of the first order, too. www.springer.com An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. 761.6 679.6 652.8 734 707.2 761.6 707.2 761.6 0 0 707.2 571.2 544 544 816 816 272 0 0 0 613.4 800 750 676.9 650 726.9 700 750 700 750 0 0 700 600 550 575 862.5 875 In this video, I introduce the Frobenius Method to solving ODEs and do a short example.Questions? >> >> P1 n=0anx. 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*}. 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 } ). 708.3 708.3 826.4 826.4 472.2 472.2 472.2 649.3 826.4 826.4 826.4 826.4 0 0 0 0 0 The Euler–Cauchy equation can be solved by taking the guess $z = u ^ { \lambda }$ with unknown parameter $\lambda \in \mathbf{C}$. Using The Frobenius Method, Find The General Solution In All Cases Of The Parameters Of The So-called Hypergeometric Equation At The Point X = 0, Given By (1 – 2)y" + [7 - (a +B+1)x]y – Aby = 0, 0,B,9 € C Check That The Solutions Are Written In Terms Of The Hypergeometric Gaus- Sian Function, Defined As F(Q.B;; 2) = (a)k(3)k 24 X 708.3 795.8 767.4 826.4 767.4 826.4 0 0 767.4 619.8 590.3 590.3 885.4 885.4 295.1 \end{equation*}, 1) $\lambda _ { 1 } = \lambda _ { 2 }$. \end{equation*}, \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 ) ! } 777.8 777.8 1000 1000 777.8 777.8 1000 777.8] 15 0 obj n; y2(x) =xr2. \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 }. The solution for $l = 0$ may contain logarithmic terms in the higher powers, starting with $( \operatorname { log } z ) z ^ { \lambda _ { 1 } }$. /LastChar 196 1277.8 811.1 811.1 875 875 666.7 666.7 666.7 666.7 666.7 666.7 888.9 888.9 888.9 endobj There is at least one Frobenius solution, in each case. https://encyclopediaofmath.org/index.php?title=Frobenius_method&oldid=50967, R. Redheffer, "Differential equations, theory and applications" , Jones and Bartlett (1991), F. Rothe, "A variant of Frobenius' method for the Emden–Fowler equation", D. Zwillinger, "Handbook of differential equations" , Acad. The other solution takes the form y2(t) = y1(t)lnt + tγ1 + 1 ∞ ∑ n = 0dntn. We classify a point x (3.6) 4. SINGULAR POINTS AND THE METHOD OF FROBENIUS 291 AseachlinearcombinationofJp(x)andJ−p(x)isasolutiontoBessel’sequationoforderp,thenas wetakethelimitaspgoeston,Yn(x)isasolutiontoBessel’sequationofordern.Italsoturnsout thatYn(x)andJn(x)arelinearlyindependent.Thereforewhennisaninteger,wehavethegeneral Formula (a1) gives the differential operator in its Frobenius normal form if $a ^ { [ N ] } ( z ) \equiv 1$. 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 761.6 489.6 endobj are $n_i$ linearly independent solutions of the differential equation (a3). /Type/Font /Name/F10 In this case, define $m_j$ to be the sum of those multiplicities for which $\lambda _ { i } - \lambda _ { j } \in \mathbf{N}$. << This page was last edited on 12 December 2020, at 22:42. 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 }. In particular, this can happen if the coe cients P(x) and Q(x) in the ODE y00+ P(x)y0+ Q(x)y = 0 fail to be de ned at a point x 0. /FontDescriptor 29 0 R 500 500 611.1 500 277.8 833.3 750 833.3 416.7 666.7 666.7 777.8 777.8 444.4 444.4 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 {\displaystyle u' (z)=\sum _ {k=0}^ {\infty } (k+r)A_ {k}z^ {k+r-1}} /Subtype/Type1 all with $\lambda = \lambda _ { 2 }$ and $l = 0 , \dots , n _ { 2 } - 1$, are $n_{2}$ linearly independent solutions of the differential equation (a3). All the three cases (Values of 'r' ) are covered in it. 295.1 826.4 501.7 501.7 826.4 795.8 752.1 767.4 811.1 722.6 693.1 833.5 795.8 382.6 Suppose the roots of the indicial equation are r 1 and r 2. 1062.5 826.4] 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 >> >> Let $1 \leq j \leq \nu$ and let $\lambda _ { i }$ be a zero of the indicial polynomial of multiplicity $n_i$ for $i = 1 , \dots , j - 1$. /FontDescriptor 14 0 R The method of Frobenius is to seek a power series solution of the form. This is usually the method we use for complicated ordinary differential equations. The method of Frobenius starts with the guess, $$\tag{a6} u ( z , \lambda ) = z ^ { \lambda } \sum _ { k = 0 } ^ { \infty } c _ { k } ( \lambda ) z ^ { k },$$, with an undetermined parameter $\lambda \in \mathbf{C}$. u ( z ) = z r ∑ k = 0 ∞ A k z k , ( A 0 ≠ 0 ) {\displaystyle u (z)=z^ {r}\sum _ {k=0}^ {\infty }A_ {k}z^ {k},\qquad (A_ {0}\neq 0)} Differentiating: u ′ ( z ) = ∑ k = 0 ∞ ( k + r ) A k z k + r − 1. /Type/Font , Here, one has to assume that $a ^ { 2_0 } \neq 0$ to obtain a regular singular point. 795.8 795.8 649.3 295.1 531.3 295.1 531.3 295.1 295.1 531.3 590.3 472.2 590.3 472.2 /FirstChar 33 /LastChar 196 also Analytic function). Press (1989). An adaption of the Frobenius method to non-linear problems is restricted to exceptional cases. • Back to Frobenius method for second solutions in three cases –n = = 0, the double root – Integer = n 0, roots differ by an integer, J-n(x) = (-1)nJ n(x) – Non-integer , easiest case, J and J- are two linearly independent solutions • General case for second solution [0,1] 2( ln() m m n with $l = 0 , \dots , n _ { j } - 1$ and $\lambda = \lambda _ { j }$, are $n_j$ linearly independent solutions of the differential equation (a3). /FontDescriptor 11 0 R In the former case there’s obviously only one Frobenius solution. 24 0 obj 935.2 351.8 611.1] This fact is the basis for the method of Frobenius. 726.9 726.9 976.9 726.9 726.9 600 300 500 300 500 300 300 500 450 450 500 450 300 << 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 734 761.6 666.2 761.6 720.6 544 707.2 734 734 1006 734 734 598.4 272 489.6 272 489.6 805.5 896.3 870.4 935.2 870.4 935.2 0 0 870.4 736.1 703.7 703.7 1055.5 1055.5 351.8 /Name/F1 /Type/Font 460.7 580.4 896 722.6 1020.4 843.3 806.2 673.6 835.7 800.2 646.2 618.6 718.8 618.8 295.1 826.4 531.3 826.4 531.3 559.7 795.8 801.4 757.3 871.7 778.7 672.4 827.9 872.8 Complications can arise if the generic assumption made above is not satisfied. 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 /Name/F5 /FirstChar 33 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 The next two theorems will enable us to develop systematic methods for finding Frobenius solutions of ( eq:7.5.2 ). P1 n=0Anx. The functions, \begin{equation*} ( \frac { \partial } { \partial \lambda } ) [ u ( z , \lambda ) ( \lambda - \lambda _ { 2 } ) ] = z ^ { \lambda_2 } + \ldots , \end{equation*}. 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 \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 ) ! } /LastChar 196 x��ZYo�6~�_�G5�fx�������d���yh{d[�ni"�q�_�U$����c�N���E�Y������(�4�����ٗ����i�Yvq�qbTV.���ɿ[�w��:��ȿo��{�XJ��7��}׷��jj?�o���UW��k�Mp��/���� The method looks simpler in the most common case of a differential operator, \tag{a9} L = a ^ { [ 2 ] } ( z ) z ^ { 2 } \left( \frac { d } { d z } \right) ^ { 2 } + a ^ { [ 1 ] } ( z ) z \left( \frac { d } { d z } \right) + a ^ { [ 0 ] } ( z ). These solutions are rational functions of$\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$. /Type/Font /FirstChar 33 The point$z = 0$is called a regular singular point of$L$. Keywords: Frobenius method; Power series method; Regular singular 1 Introduction In mathematics, the Method of Frobenius [2], named for Ferdinand Georg Frobenius, is a method to nd an in nite series solution for a second-order ordinary di erential equation of the form x2y00+p(x)y0+q(x)y= 0 … 5 See Joseph L. Neuringera, The Frobenius method for complex roots of the indicial equation, International Journal of Mathematical Education in Science and Technology, Volume 9, Issue 1, 1978, 71–77. ( \operatorname { log } z ) ^ { l } z ^ { \lambda _ { j } } + \ldots, \end{equation*}. 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]. , whatever the value of γ1 y1 ( x ) … cxe1=x, which not... 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