§10.3 Graphics

Keywords:: Bessel functions
Permalink:: http://dlmf.nist.gov/10.3

§10.3(i) Real Order and Variable
§10.3(ii) Real Order, Complex Variable
§10.3(iii) Imaginary Order, Real Variable

§10.3(i) Real Order and Variable

Notes:: These graphics were produced at NIST.
Keywords:: Bessel functions
Referenced by:: §10.74(vi)
Permalink:: http://dlmf.nist.gov/10.3.i

For the modulus and phase functions $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ , $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ , and $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ see §10.18.

Figure 10.3.1: $\mathop{J_{{0}}\/}\nolimits\!\left(x\right),\mathop{Y_{{0}}\/}\nolimits\!\left(x\right),\mathop{J_{{1}}\/}\nolimits\!\left(x\right),\mathop{Y_{{1}}\/}\nolimits\!\left(x\right),0\leq x\leq 10$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind and $x$ : real variable
Referenced by:: Graphics
Permalink:: http://dlmf.nist.gov/10.3.F1
Encodings:: pdf, png

Figure 10.3.2: $\mathop{J_{{5}}\/}\nolimits\!\left(x\right),\mathop{Y_{{5}}\/}\nolimits\!\left(x\right),\mathop{M_{{5}}\/}\nolimits\!\left(x\right),0\leq x\leq 15$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{M_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of Bessel functions and $x$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F2
Encodings:: pdf, png

Figure 10.3.3: ${\mathop{J_{{5}}\/}\nolimits^{{\prime}}}\!\left(x\right),{\mathop{Y_{{5}}\/}\nolimits^{{\prime}}}\!\left(x\right),\mathop{N_{{5}}\/}\nolimits\!\left(x\right),0\leq x\leq 15$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $\mathop{N_{{\nu}}\/}\nolimits\!\left(x\right)$ : modulus of derivatives of Bessel functions and $x$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F3
Encodings:: pdf, png

Figure 10.3.4: $\mathop{\theta _{{5}}\/}\nolimits\!\left(x\right),\mathop{\phi _{{5}}\/}\nolimits\!\left(x\right),0\leq x\leq 15$ .

Symbols:: $\mathop{\phi _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of derivatives of Bessel functions, $\mathop{\theta _{{\nu}}\/}\nolimits\!\left(x\right)$ : phase of Bessel functions and $x$ : real variable
Referenced by:: Graphics
Permalink:: http://dlmf.nist.gov/10.3.F4
Encodings:: pdf, png

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Figure 10.3.5: $\mathop{J_{{\nu}}\/}\nolimits\!\left(x\right),0\leq x\leq 10,0\leq\nu\leq 5$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: Graphics
Permalink:: http://dlmf.nist.gov/10.3.F5
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.6: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(x\right),0<x\leq 10,0\leq\nu\leq 5$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.3.F6
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.7: ${\mathop{J_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right),0\leq x\leq 10,0\leq\nu\leq 5$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $x$ : real variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.3.F7
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.8: ${\mathop{Y_{{\nu}}\/}\nolimits^{{\prime}}}\!\left(x\right),0.2\leq x\leq 10,0\leq\nu\leq 5$ .

Symbols:: $\mathop{Y_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the second kind, $x$ : real variable and $\nu$ : complex parameter
Referenced by:: Graphics
Permalink:: http://dlmf.nist.gov/10.3.F8
Encodings:: VRML, X3D, pdf, png

§10.3(ii) Real Order, Complex Variable

Notes:: These graphics were produced at NIST.
Keywords:: Hankel functions
Referenced by:: §10.26(ii), §10.74(vi)
Permalink:: http://dlmf.nist.gov/10.3.ii

In the graphics shown in this subsection, height corresponds to the absolute value of the function and color to the phase. See also p. About Color Map.

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Figure 10.3.9: $\mathop{J_{{0}}\/}\nolimits\!\left(x+iy\right),-10\leq x\leq 10,-4\leq y\leq 4$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $x$ : real variable and $y$ : real variable
Referenced by:: Graphics, Graphics
Permalink:: http://dlmf.nist.gov/10.3.F9
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.10: $\mathop{{H^{{(1)}}_{{0}}}\/}\nolimits\!\left(x+iy\right)$ , $-10\leq x\leq 5,-2.8\leq y\leq 4$ . Principal value. There is a cut along the negative real axis.

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F10
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.11: $\mathop{J_{{1}}\/}\nolimits\!\left(x+iy\right),-10\leq x\leq 10,-4\leq y\leq 4$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F11
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.12: $\mathop{{H^{{(1)}}_{{1}}}\/}\nolimits\!\left(x+iy\right)$ , $-10\leq x\leq 5,-2.8\leq y\leq 4$ . Principal value. There is a cut along the negative real axis.

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F12
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.13: $\mathop{J_{{5}}\/}\nolimits\!\left(x+iy\right),-10\leq x\leq 10,-4\leq y\leq 4$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F13
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.14: $\mathop{{H^{{(1)}}_{{5}}}\/}\nolimits\!\left(x+iy\right)$ , $-20\leq x\leq 10,-4\leq y\leq 4$ . Principal value. There is a cut along the negative real axis.

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F14
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.15: $\mathop{J_{{5.5}}\/}\nolimits\!\left(x+iy\right)$ , $-10\leq x\leq 10,-4\leq y\leq 4$ . Principal value. There is a cut along the negative real axis.

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $x$ : real variable and $y$ : real variable
Permalink:: http://dlmf.nist.gov/10.3.F15
Encodings:: VRML, X3D, pdf, png

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Figure 10.3.16: $\mathop{{H^{{(1)}}_{{5.5}}}\/}\nolimits\!\left(x+iy\right)$ , $-20\leq x\leq 10,-4\leq y\leq 4$ . Principal value. There is a cut along the negative real axis.

Symbols:: $\mathop{{H^{{(1)}}_{{\nu}}}\/}\nolimits\!\left(z\right)$ : Bessel function of the third kind (or Hankel function), $x$ : real variable and $y$ : real variable
Referenced by:: Graphics
Permalink:: http://dlmf.nist.gov/10.3.F16
Encodings:: VRML, X3D, pdf, png

§10.3(iii) Imaginary Order, Real Variable

Notes:: These graphs were produced at NIST.
Keywords:: Bessel functions
Referenced by:: §10.24
Permalink:: http://dlmf.nist.gov/10.3.iii

For the notation see §10.24.

Figure 10.3.17: $\mathop{\widetilde{J}_{{1/2}}\/}\nolimits\!\left(x\right),\mathop{\widetilde{Y}_{{1/2}}\/}\nolimits\!\left(x\right),0.01\leq x\leq 10$ .