§23.4 Graphics

Permalink:: http://dlmf.nist.gov/23.4

§23.4(i) Real Variables
§23.4(ii) Complex Variables

§23.4(i) Real Variables

Notes:: These graphs were produced at NIST.
Keywords:: Weierstrass elliptic functions, lattice
Permalink:: http://dlmf.nist.gov/23.4.i

Line graphs of the Weierstrass functions $\mathop{\wp\/}\nolimits\!\left(x\right)$ , $\mathop{\zeta\/}\nolimits\!\left(x\right)$ , and $\mathop{\sigma\/}\nolimits\!\left(x\right)$ , illustrating the lemniscatic and equianharmonic cases. (The figures in this subsection may be compared with the figures in §22.3(i).)

Figure 23.4.1: $\mathop{\wp\/}\nolimits\!\left(x;g_{2},0\right)$ for $0\leq x\leq 9$ , $g_{2}$ = 0.1, 0.2, 0.5, 0.8. (Lemniscatic case.)

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $g_{2}$ , $g_{3}$ : lattice invariants and $x$ : real part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F1
Encodings:: pdf, png

Figure 23.4.2: $\mathop{\wp\/}\nolimits\!\left(x;0,g_{3}\right)$ for $0\leq x\leq 9$ , $g_{3}$ = 0.1, 0.2, 0.5, 0.8. (Equianharmonic case.)

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $g_{2}$ , $g_{3}$ : lattice invariants and $x$ : real part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F2
Encodings:: pdf, png

Figure 23.4.3: $\mathop{\zeta\/}\nolimits\!\left(x;g_{2},0\right)$ for $0\leq x\leq 8$ , $g_{2}$ = 0.1, 0.2, 0.5, 0.8. (Lemniscatic case.)

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass zeta function, $g_{2}$ , $g_{3}$ : lattice invariants and $x$ : real part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F3
Encodings:: pdf, png

Figure 23.4.4: $\mathop{\zeta\/}\nolimits\!\left(x;0,g_{3}\right)$ for $0\leq x\leq 8$ , $g_{3}$ = 0.1, 0.2, 0.5, 0.8. (Equianharmonic case.)

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass zeta function, $g_{2}$ , $g_{3}$ : lattice invariants and $x$ : real part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F4
Encodings:: pdf, png

Figure 23.4.5: $\mathop{\sigma\/}\nolimits\!\left(x;g_{2},0\right)$ for $-5\leq x\leq 5$ , $g_{2}$ = 0.1, 0.2, 0.5, 0.8. (Lemniscatic case.)

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass sigma function, $g_{2}$ , $g_{3}$ : lattice invariants and $x$ : real part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F5
Encodings:: pdf, png

Figure 23.4.6: $\mathop{\sigma\/}\nolimits\!\left(x;0,g_{3}\right)$ for $-5\leq x\leq 5$ , $g_{3}$ = 0.1, 0.2, 0.5, 0.8. (Equianharmonic case.)

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass sigma function, $g_{2}$ , $g_{3}$ : lattice invariants and $x$ : real part of $z$
Keywords:: Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.4.F6
Encodings:: pdf, png

Figure 23.4.7: $\mathop{\wp\/}\nolimits\!\left(x\right)$ with $\omega _{{1}}=\mathop{K\/}\nolimits\!\left(k\right)$ , $\omega _{{3}}=i\!\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ for $0\leq x\leq 9$ , $k^{2}$ = 0.2, 0.8, 0.95, 0.99. (Lemniscatic case.)

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $x$ : real part of $z$ , $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
Keywords:: Weierstrass elliptic functions, lattice
Permalink:: http://dlmf.nist.gov/23.4.F7
Encodings:: pdf, png

§23.4(ii) Complex Variables

Notes:: These graphics were produced at NIST.
Keywords:: Weierstrass elliptic functions
Permalink:: http://dlmf.nist.gov/23.4.ii

Surfaces for the Weierstrass functions $\mathop{\wp\/}\nolimits\!\left(z\right)$ , $\mathop{\zeta\/}\nolimits\!\left(z\right)$ , and $\mathop{\sigma\/}\nolimits\!\left(z\right)$ . Height corresponds to the absolute value of the function and color to the phase. See also About Color Map. (The figures in this subsection may be compared with the figures in §22.3(iii).)

Visualization Help

Figure 23.4.8: $\mathop{\wp\/}\nolimits\!\left(x+iy\right)$ with $\omega _{{1}}=\mathop{K\/}\nolimits\!\left(k\right)$ , $\omega _{{3}}=i\!\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ for $-2\!\mathop{K\/}\nolimits\!\left(k\right)\leq x\leq 2\!\mathop{K\/}\nolimits\!\left(k\right)$ , $0\leq y\leq 6\!\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ , $k^{2}=0.9$ . (The scaling makes the lattice appear to be square.)

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z\right)$ (= $\mathop{\wp\/}\nolimits\!\left(z|\mathbb{L}\right)$ = $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ ): Weierstrass $\mathop{\wp\/}\nolimits$ -function, $\mathop{{K^{{\prime}}}\/}\nolimits\!\left(k\right)$ : Legendre’s complementary complete elliptic integral of the first kind, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathbb{L}$ : lattice, $x$ : real part of $z$ , $y$ : imaginary part of $z$ , $\omega _{1}$ , $\omega _{3}$ , $\omega _{2}=-\omega _{1}-\omega _{3}$ : lattice generators and $k$ : modulus
Permalink:: http://dlmf.nist.gov/23.4.F8
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 23.4.9: $\mathop{\wp\/}\nolimits\!\left(x+iy;1,4i\right)$ for $-3.8\leq x\leq 3.8$ , $-3.8\leq y\leq 3.8.$ (The variables are unscaled and the lattice is skew.)

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function, $x$ : real part of $z$ and $y$ : imaginary part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F9
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 23.4.10: $\mathop{\zeta\/}\nolimits\!\left(x+iy;1,0\right)$ for $-5\leq x\leq 5$ , $-5\leq y\leq 5.$

Symbols:: $\mathop{\zeta\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass zeta function, $x$ : real part of $z$ and $y$ : imaginary part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F10
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 23.4.11: $\mathop{\sigma\/}\nolimits\!\left(x+iy;1,i\right)$ for $-2.5\leq x\leq 2.5$ , $-2.5\leq y\leq 2.5.$

Symbols:: $\mathop{\sigma\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass sigma function, $x$ : real part of $z$ and $y$ : imaginary part of $z$
Permalink:: http://dlmf.nist.gov/23.4.F11
Encodings:: VRML, X3D, pdf, png

Visualization Help

Figure 23.4.12: $\mathop{\wp\/}\nolimits\!\left(3.7;a+ib,0\right)$ for $-5\leq a\leq 3$ , $-4\leq b\leq 4$ . There is a double zero at $a=b=0$ and double poles on the real axis.

Symbols:: $\mathop{\wp\/}\nolimits\!\left(z;g_{2},g_{3}\right)$ : Weierstrass $\mathop{\wp\/}\nolimits$ -function
Permalink:: http://dlmf.nist.gov/23.4.F12
Encodings:: VRML, X3D, pdf, png

§23.4 Graphics

Contents

§23.4(i) Real Variables

§23.4(ii) Complex Variables