{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:36:25Z","timestamp":1776728185161,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"235","license":[{"start":{"date-parts":[[2001,6,12]],"date-time":"2001-06-12T00:00:00Z","timestamp":992304000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Bounds for the distance\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"StartAbsoluteValue c Subscript nu comma s Baseline minus c Subscript nu plus-or-minus 1 comma s prime Baseline EndAbsoluteValue\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>s<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  \u00b1\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>s<\/mml:mi>\n                                  <mml:mo>\u2032<\/mml:mo>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">|c_{\\nu ,s}-c_{\\nu \\pm 1 , s\u2019 }|<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    between adjacent zeros of cylinder functions are given;\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s\">\n                        <mml:semantics>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">s<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"s prime\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>s<\/mml:mi>\n                            <mml:mo>\u2032<\/mml:mo>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">s\u2019<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"there-does-not-exist c Subscript nu comma s Sub Superscript double-prime Subscript Baseline element-of right-bracket c Subscript nu comma s Baseline comma c Subscript nu plus-or-minus 1 comma s Sub Superscript prime Subscript Baseline left-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u2204\n                              \n                            <\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>s<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">\n                                      \u2032\n                                      \n                                    <\/mml:mi>\n                                    <mml:mi class=\"MJX-variant\" mathvariant=\"normal\">\n                                      \u2032\n                                      \n                                    <\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>s<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  \u00b1\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:msup>\n                                  <mml:mi>s<\/mml:mi>\n                                  <mml:mo>\u2032<\/mml:mo>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\nexists c_{\\nu ,s^{\\prime \\prime }}\\in \\ ]c_{\\nu ,s},c_{\\nu \\pm 1,s\u2019 }[<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ;\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript nu comma k\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>c<\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03bd\n                                \n                              <\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">c_{\\nu ,k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    stands for the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    th positive zero of the cylinder (Bessel) function\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper C Subscript nu Baseline left-parenthesis x right-parenthesis equals cosine alpha upper J Subscript nu Baseline left-parenthesis x right-parenthesis minus sine alpha upper Y Subscript nu Baseline left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>cos<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>J<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>sin<\/mml:mi>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:msub>\n                              <mml:mi>Y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {C}_{\\nu }(x)=\\cos \\alpha J_{\\nu }(x) - \\sin \\alpha Y_{\\nu }(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha element-of left-bracket 0 comma pi left-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03b1\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\n                              \u03c0\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha \\in [0,\\pi [<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"nu element-of double-struck upper R\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u03bd\n                              \n                            <\/mml:mi>\n                            <mml:mo>\n                              \u2208\n                              \n                            <\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\nu \\in {\\mathbb R}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . These bounds, together with the application of modified (global) Newton methods based on the monotonic functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f Subscript nu Baseline left-parenthesis x right-parenthesis equals x Superscript 2 nu minus 1 Baseline script upper C Subscript nu Baseline left-parenthesis x right-parenthesis slash script upper C Subscript nu minus 1 Baseline left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>f<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f_{\\nu }(x)=x^{2\\nu -1}\\mathcal {C}_{\\nu }(x)\/\\mathcal {C}_{\\nu -1}(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"g Subscript nu Baseline left-parenthesis x right-parenthesis equals minus x Superscript minus left-parenthesis 2 nu plus 1 right-parenthesis Baseline script upper C Subscript nu Baseline left-parenthesis x right-parenthesis slash script upper C Subscript nu plus 1 Baseline left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>g<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mo>\n                                  \u2212\n                                  \n                                <\/mml:mo>\n                                <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">g_{\\nu }(x)=-x^{-(2\\nu +1)}\\mathcal {C}_{\\nu }(x)\/\\mathcal {C}_{\\nu +1}(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , give rise to forward (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript nu comma k Baseline right-arrow c Subscript nu comma k plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_{\\nu ,k} \\rightarrow c_{\\nu ,k+1}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) and backward (\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"c Subscript nu comma k plus 1 Baseline right-arrow c Subscript nu comma k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">\n                              \u2192\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>c<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">c_{\\nu ,k+1} \\rightarrow c_{\\nu ,k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) iterative relations between consecutive zeros of cylinder functions. The problem of finding all the positive real zeros of Bessel functions\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script upper C Subscript nu Baseline left-parenthesis x right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi class=\"MJX-tex-caligraphic\" mathvariant=\"script\">C<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>\n                                  \u03bd\n                                  \n                                <\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathcal {C}_{\\nu }(x)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for any real\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"alpha\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b1\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\alpha<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"nu\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03bd\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\nu<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    inside an interval\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-bracket x 1 comma x 2 right-bracket\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">[<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">]<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">[x_{1},x_{2}]<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"x 1 greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">x_{1}&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , is solved in a simple way.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01243-6","type":"journal-article","created":{"date-parts":[[2002,11,6]],"date-time":"2002-11-06T14:02:22Z","timestamp":1036591342000},"page":"1205-1220","source":"Crossref","is-referenced-by-count":13,"title":["Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros"],"prefix":"10.1090","volume":"70","author":[{"given":"Javier","family":"Segura","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,6,12]]},"reference":[{"key":"1","isbn-type":"print","volume-title":"Handbook of mathematical functions with formulas, graphs, and mathematical tables","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0486612724"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1080\/02522667.1996.10699281","article-title":"On asymptotic formulas for the first zero of the Bessel function \ud835\udc3d_{\ud835\udf08}","volume":"17","author":"Belingeri, Carlo","year":"1996","journal-title":"J. Inform. Optim. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0252-2667","issn-type":"print"},{"issue":"158","key":"3","doi-asserted-by":"publisher","first-page":"589","DOI":"10.2307\/2007292","article-title":"An upper bound for the first zero of Bessel functions","volume":"38","author":"Chambers, Ll. G.","year":"1982","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"4","doi-asserted-by":"publisher","first-page":"614","DOI":"10.1137\/0516045","article-title":"On the convexity of the zeros of Bessel functions","volume":"16","author":"Elbert, \u00c1rp\u00e1d","year":"1985","journal-title":"SIAM J. Math. 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Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1331-4343","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/0377-0427(92)90055-3","article-title":"A differential inequality for the positive zeros of Bessel functions","volume":"44","author":"Ifantis, E. K.","year":"1992","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"8","unstructured":"K.S. K\u00f6lbig, CERNLIB-Short Writeups, subroutine DBZEJY(C345), library MATHLIB http:\/\/wwwcn.cern.ch\/shortwrupsdir\/index.html."},{"key":"9","doi-asserted-by":"crossref","unstructured":"P. Kravanja, O. Ragos, M. N. Vrahatis, F.A. Zafiropoulos. \u201cZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument\u201d. Comput. Phys. Commun. 113 (1998) 220-238.","DOI":"10.1016\/S0010-4655(98)00064-2"},{"key":"10","volume-title":"Special functions and their applications","author":"Lebedev, N. N.","year":"1972"},{"key":"11","volume-title":"Mathematical functions and their approximations","author":"Luke, Yudell L.","year":"1975"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1016\/S0377-0427(96)00065-9","article-title":"\u201cBest possible\u201d upper bounds for the first positive zeros of Bessel functions\u2014the finite part","volume":"75","author":"Lorch, Lee","year":"1996","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"13","doi-asserted-by":"crossref","unstructured":"M. E. Muldoon, \u201cConvexity properties of special functions and their zeros\u201d in Recent Progress in Inequalities, pp. 309-323, Kluwer Academics Publishers, Dordrecht-Boston-London (1998).","DOI":"10.1007\/978-94-015-9086-0_17"},{"key":"14","series-title":"Royal Society Mathematical Tables, Vol. 7","volume-title":"Bessel functions. Part III: Zeros and associated values","year":"1960"},{"key":"15","doi-asserted-by":"crossref","unstructured":"R. Piessens, \u201cRational approximations for the zeros of Bessel functions\u201d, J. Comput. Phys. 42 (1981) 403-405.","DOI":"10.1016\/0021-9991(81)90253-9"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"188","DOI":"10.1016\/0021-9991(84)90060-3","article-title":"Chebyshev series approximations for the zeros of the Bessel functions","volume":"53","author":"Piessens, R.","year":"1984","journal-title":"J. Comput. 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Gr\\`ece (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0373-1391","issn-type":"print"},{"key":"19","isbn-type":"print","volume-title":"Numerical recipes in FORTRAN","author":"Press, William H.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/052143064X","edition":"2"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/S0377-0427(98)00088-0","article-title":"Computational matrix representation modules for linear operators with explicit constructions for a class of Lie operators","volume":"98","author":"Fu, Jyun-Horng","year":"1998","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","unstructured":"J. Segura & A. Gil, \u201cELF and GNOME: Two tiny codes to evaluate the real zeros of the Bessel functions of the first kind for real orders\", Comput. Phys. 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