§ 2.7. Differential Equations

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§2.7(i)Regular Singularities: Fuchs-Frobenius Theory
§2.7(ii)Irregular Singularities of Rank 1
§2.7(iii)Liouville-Green (WKBJ) Approximation
§2.7(iv)Numerically Satisfactory Solutions

§ 2.7(i). Regular Singularities: Fuchs-Frobenius Theory

Notes:: See Olver (1997b, pp. 148–152).
Permalink:: http://dlmf.nist.gov/2.7.SS1

An ordinary point of the differential equation

2.7.1 $\frac{{d}^{2}w}{{dz}^{2}}+f(z)\frac{dw}{dz}+g(z)w=0$

Defines:: $w$ : DE solution, $f(z)$ : analytic function and $g(z)$ : analytic function
Referenced by:: §2.11(v), §2.7(i), §2.7(i)
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is one at which the coefficients $f(z)$ and $g(z)$ are analytic. All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients.

Other points $z_{0}$ are singularities of the differential equation. If both $(z-z_{0})f(z)$ and $(z-z_{0})^{2}g(z)$ are analytic at $z_{0}$ , then $z_{0}$ is a regular singularity (or singularity of the first kind). All other singularities are classified as irregular.

In a punctured neighborhood $\mathbf{N}$ of a regular singularity $z_{0}$

2.7.2

$f(z)=\sum _{{s=0}}^{{\infty}}f_{s}(z-z_{0})^{{s-1}},$

$g(z)=\sum _{{s=0}}^{{\infty}}g_{s}(z-z_{0})^{{s-2}},$

Defines:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
Symbols:: $f(z)$ : analytic function and $g(z)$ : analytic function
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with at least one of the coefficients $f_{0}$ , $g_{0}$ , $g_{1}$ nonzero. Let $\alpha _{1}$ , $\alpha _{2}$ denote the indices or exponents, that is, the roots of the indicial equation

2.7.3 $Q(\alpha)\equiv\alpha(\alpha-1)+f_{0}\alpha+g_{0}=0.$

Defines:: $f_{s}$ : coefficients, $g_{s}$ : coefficients and $Q(\alpha)$ : indicial function
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Provided that $\alpha _{1}-\alpha _{2}$ is not zero or an integer, equation (2.7.1) has independent solutions $w_{j}(z)$ , $j=1,2$ , such that

2.7.4 $w_{j}(z)=(z-z_{0})^{{\alpha _{j}}}\sum _{{s=0}}^{{\infty}}a_{{s,j}}(z-z_{0})^{s},$ $z\in\mathbf{N}$ ,

Defines:: $\mathbf{N}$ : punctured neighborhood, $w_{j}(z)$ : solutions and $a_{n}$ : coefficients
Referenced by:: §2.7(i), §2.7(i)
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with $a_{{0,j}}=1$ , and

2.7.5 $Q(\alpha _{j}+s)a_{{s,j}}=-\sum _{{r=0}}^{{s-1}}\left((\alpha _{j}+r)f_{{s-r}}+g_{{s-r}}\right)a_{{r,j}},$

Defines:: $f_{s}$ : coefficients, $g_{s}$ : coefficients, $Q(\alpha)$ : indicial function and $a_{n}$ : coefficients
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when $s=1,2,3,\dots$ .

If $\alpha _{1}-\alpha _{2}=0,1,2,\dots$ , then (2.7.4) applies only in the case $j=1$ . But there is an independent solution

2.7.6 $w_{2}(z)=(z-z_{0})^{{\alpha _{2}}}\sum _{{\substack{s=0\\ s\neq\alpha _{1}-\alpha _{2}}}}^{{\infty}}b_{s}(z-z_{0})^{s}+cw_{1}(z)\ln\!\left(z-z_{0}\right),$ $z\in\mathbf{N}$ .

Defines:: $b_{s}$ : coefficients and $c$ : constant
Symbols:: $\mathbf{N}$ : punctured neighborhood and $w_{j}(z)$ : solutions
Referenced by:: §2.7(i), §2.9(ii)
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The coefficients $b_{s}$ and constant $c$ are again determined by equating coefficients in the differential equation, beginning with $c=1$ when $\alpha _{1}-\alpha _{2}=0$ , or with $b_{0}=1$ when $\alpha _{1}-\alpha _{2}=1,2,3,\dots$ .

The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from $z_{0}$ .

To include the point at infinity in the foregoing classification scheme, we transform it into the origin by replacing $z$ in (2.7.1) with $1/z$ ; see Olver (1997b, pp. 153–154). For corresponding definitions, together with examples, for linear differential equations of arbitrary order see §§Ch.16 – Ch.16.

§ 2.7(ii). Irregular Singularities of Rank 1

Notes:: See Olver (1997b, pp. 229–236), Olver (1994a), Olde Daalhuis and Olver (1994), and Olde Daalhuis (1998a).
Referenced by:: §2.11(v), §2.9(i)
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If the singularities of $f(z)$ and $g(z)$ at $z_{0}$ are no worse than poles, then $z_{0}$ has rank $\ell-1$ , where $\ell$ is the least integer such that $(z-z_{0})^{{\ell}}f(z)$ and $(z-z_{0})^{{2\ell}}g(z)$ are analytic at $z_{0}$ . Thus a regular singularity has rank 0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. Then

2.7.7

$f(z)=\sum _{{s=0}}^{{\infty}}\frac{f_{s}}{z^{s}},$

$g(z)=\sum _{{s=0}}^{{\infty}}\frac{g_{s}}{z^{s}},$

Symbols:: $f(z)$ : analytic function, $g(z)$ : analytic function, $f_{s}$ : coefficients and $g_{s}$ : coefficients
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these series converging in an annulus $|z|>a$ , with at least one of $f_{0}$ , $g_{0}$ , $g_{1}$ nonzero.

Formal solutions are

2.7.8 $e^{{\lambda _{j}z}}z^{{\mu _{j}}}\sum _{{s=0}}^{{\infty}}\frac{a_{{s,j}}}{z^{s}},$ $j=1,2$ ,

Defines:: $a_{{s,j}}$ : coefficients, $\lambda _{j}$ : roots and $\mu _{j}$ : quantities
Referenced by:: §2.11(v), §2.7(ii)
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where $\lambda _{1}$ , $\lambda _{2}$ are the roots of the characteristic equation

2.7.9 $\lambda^{2}+f_{0}\lambda+g_{0}=0,$

Symbols:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
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2.7.10 $\mu _{j}=-(f_{1}\lambda _{j}+g_{1})/(f_{0}+2\lambda _{j}),$

Defines:: $\lambda _{j}$ : roots and $\mu _{j}$ : quantities
Symbols:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
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$a_{{0,j}}=1$ , and

2.7.11 $(f_{0}+2\lambda _{j})sa_{{s,j}}=(s-\mu _{j})(s-1-\mu _{j})a_{{s-1,j}}+\sum _{{r=1}}^{{s}}\left(\lambda _{j}f_{{r+1}}+g_{{r+1}}-(s-r-\mu _{j})f_{r}\right)a_{{s-r,j}},$

Defines:: $a_{{s,j}}$ : coefficients, $\lambda _{j}$ : roots and $\mu _{j}$ : quantities
Symbols:: $f_{s}$ : coefficients and $g_{s}$ : coefficients
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when $s=1,2,\dots$ . The construction fails iff $\lambda _{1}=\lambda _{2}$ , that is, when $f_{0}^{2}=4g_{0}$ : this case is treated below.

For large $s$ ,

2.7.12 $a_{{s,1}}\sim\frac{\Lambda _{1}}{(\lambda _{1}-\lambda _{2})^{s}}\*\sum _{{j=0}}^{{\infty}}{a_{{j,2}}(\lambda _{1}-\lambda _{2})^{j}\Gamma\!\left(s+\mu _{{2}}-\mu _{1}-j\right)},$

Defines:: $\Lambda _{j}$ : constants
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $a_{{s,j}}$ : coefficients, $\lambda _{j}$ : roots and $\mu _{j}$ : quantities
Referenced by:: §2.11(v), §2.7(ii)
Permalink:: http://dlmf.nist.gov/2.7.E12
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2.7.13 $a_{{s,2}}\sim\frac{\Lambda _{2}}{(\lambda _{2}-\lambda _{1})^{s}}\*\sum _{{j=0}}^{{\infty}}{a_{{j,1}}(\lambda _{2}-\lambda _{1})^{j}\Gamma\!\left(s+\mu _{1}-\mu _{2}-j\right)},$

Defines:: $\Lambda _{j}$ : constants
Symbols:: $\Gamma\!\left(z\right)$ : Gamma function, $\sim$ : asymptotically equal, $a_{{s,j}}$ : coefficients, $\lambda _{j}$ : roots and $\mu _{j}$ : quantities
Referenced by:: §2.11(v), §2.7(ii)
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where $\Lambda _{1}$ and $\Lambda _{2}$ are constants, and the $J$ th remainder terms in the sums are $O\!\left(\Gamma\!\left(s+\mu _{2}-\mu _{1}-J\right)\right)$ and $O\!\left(\Gamma\!\left(s+\mu _{1}-\mu _{2}-J\right)\right)$ , respectively (Olver (1994a)). Hence unless the series (2.7.8) terminate (in which case the corresponding $\Lambda _{j}$ is zero) they diverge. However, there are unique and linearly independent solutions $w_{j}(z)$ , $j=1,2$ , such that

2.7.14 $w_{j}(z)\sim e^{{\lambda _{j}z}}((\lambda _{2}-\lambda _{1})z)^{{\mu _{j}}}\sum _{{s=0}}^{{\infty}}\frac{a_{{s,j}}}{z^{s}}$

Symbols:: $\sim$ : asymptotically equal, $a_{{s,j}}$ : coefficients, $\lambda _{j}$ : roots, $\mu _{j}$ : quantities and $w_{j}(z)$ : solutions
Referenced by:: §2.11(v), §2.7(ii), §2.7(ii)
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as $z\to\infty$ in the sectors

2.7.15 $-\tfrac{3}{2}\pi+\delta\leq\mathrm{ph}\!\left((\lambda _{2}-\lambda _{1})z\right)\leq\tfrac{3}{2}\pi-\delta,$ $j=1$ ,

Defines:: $\delta$ : arbitrary small positive constant
Symbols:: $\lambda _{j}$ : roots
Referenced by:: §2.7(ii)
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2.7.16 $-\tfrac{1}{2}\pi+\delta\leq\mathrm{ph}\!\left((\lambda _{2}-\lambda _{1})z\right)\leq\tfrac{5}{2}\pi-\delta,$ $j=2$ ,

Defines:: $\delta$ : arbitrary small positive constant
Symbols:: $\lambda _{j}$ : roots
Referenced by:: §2.7(ii)
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$\delta$ being an arbitrary small positive constant.

Although the expansions (2.7.14) apply only in the sectors (2.7.15) and (2.7.16), each solution $w_{j}(z)$ can be continued analytically into any other sector. Typical connection formulas are

2.7.17

$w_{1}(z)=e^{{2\pi i\mu _{1}}}w_{1}(ze^{{-2\pi i}})+C_{1}w_{2}(z),$

$w_{2}(z)=e^{{-2\pi i\mu _{2}}}w_{2}(ze^{{2\pi i}})+C_{2}w_{1}(z),$

Defines:: $C_{1}$ , $C_{2}$ : Stokes multipliers
Symbols:: $\mu _{j}$ : quantities and $w_{j}(z)$ : solutions
Referenced by:: §2.11(v)
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in which $C_{1}$ , $C_{2}$ are constants, the so-called Stokes multipliers. In combination with (2.7.14) these formulas yield asymptotic expansions for $w_{1}(z)$ in $\frac{1}{2}\pi+\delta\leq\mathrm{ph}\!\left((\lambda _{2}-\lambda _{1})z\right)\leq\frac{5}{2}\pi-\delta$ , and $w_{2}(z)$ in $-\frac{3}{2}\pi+\delta\leq\mathrm{ph}\!\left((\lambda _{2}-\lambda _{1})z\right)\leq\frac{1}{2}\pi-\delta$ . Furthermore,

2.7.18

$\Lambda _{1}=-ie^{{(\mu _{2}-\mu _{1})\pi i}}C_{1}/(2\pi),$

$\Lambda _{2}=iC_{2}/(2\pi).$

Defines:: $C_{1}$ , $C_{2}$ : Stokes multipliers
Symbols:: $\mu _{j}$ : quantities and $\Lambda _{j}$ : constants
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Note that the coefficients in the expansions (2.7.12), (2.7.13) for the “late” coefficients, that is, $a_{{s,1}}$ , $a_{{s,2}}$ with $s$ large, are the “early” coefficients $a_{{j,2}}$ , $a_{{j,1}}$ with $j$ small. This phenomenon is an example of resurgence, a classification due to Écalle (1981a, b). See §2.11(v) for other examples.

The exceptional case $f_{0}^{2}=4g_{0}$ is handled by Fabry's transformation:

2.7.19

$w=e^{{-f_{0}z/2}}W,$

$t=z^{{1/2}}.$

Symbols:: $w$ : DE solution and $f_{s}$ : coefficients
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The transformed differential equation either has a regular singularity at $t=\infty$ , or its characteristic equation has unequal roots.

For error bounds for (2.7.14) see Olver (1997b, Chapter 7). For the calculation of Stokes multipliers see Olde Daalhuis and Olver (1995b). For extensions to singularities of higher rank see Olver and Stenger (1965). For extensions to higher-order differential equations see Stenger (1966a, b), Olver (1997a), and Olde Daalhuis and Olver (1998).

§ 2.7(iii). Liouville-Green (WKBJ) Approximation

Notes:: See Olver (1997b, pp. 190–200).
Referenced by:: §2.9(i), §2.9(iii)
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For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:

¶ Liouville-Green Approximation Theorem

In a finite or infinite interval $(a_{1},a_{2})$ let $f(x)$ be real, positive, and twice-continuously differentiable, and $g(x)$ be continuous. Then in $(a_{1},a_{2})$ the differential equation

2.7.20 $\frac{{d}^{2}w}{{dx}^{2}}=(f(x)+g(x))w$

Defines:: $f(z)$ : real positive analytic function and $g(z)$ : analytic function
Symbols:: $w$ : DE solution
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has twice-continuously differentiable solutions

2.7.21 $w_{1}(x)=f^{{-1/4}}(x)\exp\!\left(\int f^{{1/2}}(x)dx\right)\*\left(1+\epsilon _{1}(x)\right),$

Defines:: $f(z)$ : real positive analytic function and $\epsilon _{j}(x)$ : function
Symbols:: $w_{j}(z)$ : solutions
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2.7.22 $w_{2}(x)=f^{{-1/4}}(x)\exp\!\left(-\int f^{{1/2}}(x)dx\right)\*\left(1+\epsilon _{2}(x)\right),$

Defines:: $f(z)$ : real positive analytic function and $\epsilon _{j}(x)$ : function
Symbols:: $w_{j}(z)$ : solutions
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such that

2.7.23 $|\epsilon _{j}(x)|,\;\;\tfrac{1}{2}f^{{-1/2}}(x)|\epsilon _{j}^{{\prime}}(x)|\leq\exp\!\left(\tfrac{1}{2}\mathcal{V}_{{a_{j},x}}\!\left(F\right)\right)-1,$ $j=1,2$ ,

Defines:: $(a_{1},a_{2})$ : interval, $f(z)$ : real positive analytic function, $\epsilon _{j}(x)$ : function and $F$ : error-control function
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provided that $\mathcal{V}_{{a_{j},x}}\!\left(F\right)<\infty$ . Here $F(x)$ is the error-control function

2.7.24 $F(x)=\int\left(\frac{1}{f^{{1/4}}}\frac{{d}^{2}}{{dx}^{2}}\left(\frac{1}{f^{{1/4}}}\right)-\frac{g}{f^{{1/2}}}\right)dx,$

Defines:: $f(z)$ : real positive analytic function, $g(z)$ : analytic function and $F$ : error-control function
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and $\mathcal{V}$ denotes the variational operator; thus

2.7.25 $\mathcal{V}_{{a_{j},x}}\!\left(F\right)=\int _{{a_{j}}}^{{x}}\left|\left(\frac{1}{f^{{1/4}}(t)}\frac{{d}^{2}}{{dt}^{2}}\left(\frac{1}{f^{{1/4}}(t)}\right)-\frac{g(t)}{f^{{1/2}}(t)}\right)dt\right|.$

Defines:: $(a_{1},a_{2})$ : interval, $f(z)$ : real positive analytic function, $g(z)$ : analytic function and $F$ : error-control function
Permalink:: http://dlmf.nist.gov/2.7.E25
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If $\mathcal{V}_{{a_{1},a_{2}}}\!\left(F\right)<\infty$ , then

2.7.26 $w_{1}(x)\sim f^{{-1/4}}(x)\exp\!\left(\int f^{{1/2}}(x)dx\right),$ $x\to a_{1}+$ ,

Symbols:: $\sim$ : asymptotically equal, $(a_{1},a_{2})$ : interval, $f(z)$ : real positive analytic function and $w_{j}(z)$ : solutions
Referenced by:: §2.7(iii)
Permalink:: http://dlmf.nist.gov/2.7.E26
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2.7.27 $w_{2}(x)\sim f^{{-1/4}}(x)\exp\!\left(-\int f^{{1/2}}(x)dx\right),$ $x\to a_{2}-$ .

Symbols:: $\sim$ : asymptotically equal, $(a_{1},a_{2})$ : interval, $f(z)$ : real positive analytic function and $w_{j}(z)$ : solutions
Referenced by:: §2.7(iii)
Permalink:: http://dlmf.nist.gov/2.7.E27
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Suppose in addition $|\int f^{{1/2}}(x)dx|$ is unbounded as $x\to a_{1}+$ and $x\to a_{2}-$ . Then there are solutions $w_{3}(x)$ , $w_{4}(x)$ , such that

2.7.28 $w_{3}(x)\sim f^{{-1/4}}(x)\exp\!\left(\int f^{{1/2}}(x)dx\right),$ $x\to a_{2}-$ ,

Symbols:: $\sim$ : asymptotically equal, $(a_{1},a_{2})$ : interval, $f(z)$ : real positive analytic function and $w_{j}(z)$ : solutions
Referenced by:: §2.7(iii)
Permalink:: http://dlmf.nist.gov/2.7.E28
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2.7.29 $w_{4}(x)\sim f^{{-1/4}}(x)\exp\!\left(-\int f^{{1/2}}(x)dx\right),$ $x\to a_{1}+$ .

Symbols:: $\sim$ : asymptotically equal, $(a_{1},a_{2})$ : interval, $f(z)$ : real positive analytic function and $w_{j}(z)$ : solutions
Referenced by:: §2.7(iii)
Permalink:: http://dlmf.nist.gov/2.7.E29
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The solutions with the properties (2.7.26), (2.7.27) are unique, but not those with the properties (2.7.28), (2.7.29). In fact, since

2.7.30 $w_{1}(x)/w_{4}(x)\to 0,$ $x\to a_{1}+$ ,

Symbols:: $(a_{1},a_{2})$ : interval and $w_{j}(z)$ : solutions
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$w_{1}(x)$ is a recessive (or subdominant) solution as $x\to a_{1}+$ , and $w_{4}(x)$ is a dominant solution as $x\to a_{1}+$ . Similarly for $w_{2}(x)$ and $w_{3}(x)$ as $x\to a_{2}-$ .

¶ Example

2.7.31 $\frac{{d}^{2}w}{{dx}^{2}}=(x+\ln x)w,$ $0<x<\infty$ .

Symbols:: $w$ : DE solution
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We cannot take $f=x$ and $g=\ln x$ because $\int gf^{{-1/2}}dx$ would diverge as $x\to+\infty$ . Instead set $f=x+\ln x$ , $g=0$ . By approximating

2.7.32 $f^{{1/2}}=x^{{1/2}}+\tfrac{1}{2}x^{{-1/2}}\ln x+O\!\left(x^{{-3/2}}(\ln x)^{2}\right),$

Symbols:: $O\!\left(x\right)$ : order symbol
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we arrive at

2.7.33 $w_{2}(x)\sim x^{{-(1/4)-\sqrt{x}}}\exp\!\left(2x^{{1/2}}-\tfrac{2}{3}x^{{3/2}}\right),$

Symbols:: $\sim$ : asymptotically equal and $w_{j}(z)$ : solutions
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2.7.34 $w_{3}(x)\sim x^{{-(1/4)+\sqrt{x}}}\exp\!\left(\tfrac{2}{3}x^{{3/2}}-2x^{{1/2}}\right),$

Symbols:: $\sim$ : asymptotically equal and $w_{j}(z)$ : solutions
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as $x\to+\infty$ , $w_{2}(x)$ being recessive and $w_{3}(x)$ dominant.

For other examples, and also the corresponding results when $f(x)$ is negative, see Olver (1997b, Chapter 6), Olver (1980), Taylor (1978, 1982), and Smith (1986). The first of these references includes extensions to complex variables and reversions for zeros.

§ 2.7(iv). Numerically Satisfactory Solutions

Notes:: See Olver (1997b, pp. 154–155).
Referenced by:: §2.9(i), §9.2(iii)
Permalink:: http://dlmf.nist.gov/2.7.SS4

One pair of independent solutions of the equation

2.7.35 $\ifrac{{d}^{2}w}{{dz}^{2}}=w$

Symbols:: $w$ : DE solution
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is $w_{1}(z)=e^{z}$ , $w_{2}(z)=e^{{-z}}$ . Another is $w_{3}(z)=\cosh z$ , $w_{4}(z)=\sinh z$ . In theory either pair may be used to construct any other solution

2.7.36 $w(z)=Aw_{1}(z)+Bw_{2}(z),$

Defines:: $A$ : constant and $B$ : constant
Symbols:: $w$ : DE solution and $w_{j}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/2.7.E36
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2.7.37 $w(z)=Cw_{3}(z)+Dw_{4}(z),$

Defines:: $C$ : constant and $D$ : constant
Symbols:: $w$ : DE solution and $w_{j}(z)$ : solutions
Permalink:: http://dlmf.nist.gov/2.7.E37
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where $A,B,C,D$ are constants. From the numerical standpoint, however, the pair $w_{3}(z)$ and $w_{4}(z)$ has the drawback that severe numerical cancellation can occur with certain combinations of $C$ and $D$ , for example if $C$ and $D$ are equal, or nearly equal, and $z$ , or $\realpart{z}$ , is large and negative. This kind of cancellation cannot take place with $w_{1}(z)$ and $w_{2}(z)$ , and for this reason, and following Miller (1950), we call $w_{1}(z)$ and $w_{2}(z)$ a numerically satisfactory pair of solutions.

The solutions $w_{1}(z)$ and $w_{2}(z)$ are respectively recessive and dominant as $\realpart{z}\to-\infty$ , and vice versa as $\realpart{z}\to+\infty$ . This is characteristic of numerically satisfactory pairs. In a neighborhood, or sectorial neighborhood of a singularity, one member has to be recessive. In consequence, if a differential equation has more than one singularity in the extended plane, then usually more than two standard solutions need to be chosen in order to have numerically satisfactory representations everywhere.

In oscillatory intervals, and again following Miller (1950), we call a pair of solutions numerically satisfactory if asymptotically they have the same amplitude and are $\tfrac{1}{2}\pi$ out of phase.