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1: 19.11 Addition Theorems

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19.11.1 $$F(\theta ,k)+F(\varphi ,k)=F(\psi ,k),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Referenced by:

§19.11(i)

Permalink:

http://dlmf.nist.gov/19.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.2 $$E(\theta ,k)+E(\varphi ,k)=E(\psi ,k)+k^2\sin \theta \sin \varphi \sin \psi .$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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19.11.7 $$F(\varphi ,k)=K\left(k\right)-F(\theta ,k),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus and $\theta$: amplitude

Permalink:

http://dlmf.nist.gov/19.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(ii), §19.11 and Ch.19

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19.11.12 $$F(\psi ,k)=2F(\theta ,k),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $k$: real or complex modulus, $\theta$: amplitude and $\psi$: angle

Permalink:

http://dlmf.nist.gov/19.11.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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19.11.14 $$\sin \psi =(\sin 2\theta )\mathrm{\Delta }(\theta )/(1-k^2{\sin }^4\theta ),$$

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Defines:

$\psi$: angle (locally)

Symbols:

$\sin z$: sine function, $k$: real or complex modulus, $\theta$: amplitude and $\mathrm{\Delta }(\theta )$

Permalink:

http://dlmf.nist.gov/19.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.11(iii), §19.11 and Ch.19

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2: 19.4 Derivatives and Differential Equations

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19.4.3 $$\frac{d^{2}E\left(k\right)}{{dk}^2}=-\frac{1}{k}\frac{dK\left(k\right)}{dk}=\frac{k_{}^{\prime }{}_{}{}^{2}K\left(k\right)-E\left(k\right)}{k^{2}k_{}^{\prime }{}_{}{}^2},$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.5 $$\frac{\partial F(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-k_{}^{\prime }{}_{}{}^{2}F(\varphi ,k)}{kk_{}^{\prime }{}_{}{}^2}-\frac{k\sin \varphi \cos \varphi }{k_{}^{\prime }{}_{}{}^2\sqrt{1-k^2{\sin }^2\varphi }},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.6 $$\frac{\partial E(\varphi ,k)}{\partial k}=\frac{E(\varphi ,k)-F(\varphi ,k)}{k},$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.30(i)

Permalink:

http://dlmf.nist.gov/19.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(i), §19.4 and Ch.19

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19.4.8 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+(1-3k^2)D_k-k)F(\varphi ,k)=\frac{-k\sin \varphi \cos \varphi }{{(1-k^2{\sin }^2\varphi )}^{3/2}},$$

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Symbols:

$\cos z$: cosine function, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Referenced by:

§19.4(ii)

Permalink:

http://dlmf.nist.gov/19.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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19.4.9 $$(kk_{}^{\prime }{}_{}{}^2{D}_k^2+k_{}^{\prime }{}_{}{}^2{D}_k+k)E(\varphi ,k)=\frac{k\sin \varphi \cos \varphi }{\sqrt{1-k^2{\sin }^2\varphi }}.$$

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Symbols:

$\cos z$: cosine function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and $D_k$: differential operator

Permalink:

http://dlmf.nist.gov/19.4.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.4(ii), §19.4 and Ch.19

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3: 20.12 Mathematical Applications

… ►The space of complex tori $ℂ/(\mathbb{Z}+\tau \mathbb{Z})$ (that is, the set of complex numbers $z$ in which two of these numbers $z_1$ and $z_2$ are regarded as equivalent if there exist integers $m,n$ such that $z_1-z_2=m+\tau n$) is mapped into the projective space $P^3$ via the identification $z\to ({\theta }_1\left(2z\right|\tau ),{\theta }_2\left(2z\right|\tau ),{\theta }_3\left(2z\right|\tau ),{\theta }_4\left(2z\right|\tau ))$. …

4: 19.9 Inequalities

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19.9.2

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\ln z$: principal branch of logarithm function, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Referenced by:

§19.9(i), §19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

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19.9.9 $$L(a,b)=4aE\left(k\right),$$ $k^2=1-(b^2/a^2)$, $a>b$.

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Symbols:

$E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $k$: real or complex modulus and $L(a,b)$: perimeter

Referenced by:

§19.9(i)

Permalink:

http://dlmf.nist.gov/19.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(i), §19.9 and Ch.19

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19.9.11 $$\varphi \le F(\varphi ,k)\le \min (\varphi /\mathrm{\Delta },{\mathrm{gd}}^{-1}\left(\varphi \right)),$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, ${\mathrm{gd}}^{-1}\left(x\right)$: inverse Gudermannian function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.12 $$\max (\sin \varphi ,\varphi \mathrm{\Delta })\le E(\varphi ,k)\le \varphi ,$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and $\mathrm{\Delta }$

Permalink:

http://dlmf.nist.gov/19.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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19.9.17 $$L\le F(\varphi ,k)\le \sqrt{UL}\le \frac{1}{2}(U+L)\le U,$$

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Symbols:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $L$: lower limit and $U$: upper limit

Permalink:

http://dlmf.nist.gov/19.9.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.9(ii), §19.9 and Ch.19

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5: 19.5 Maclaurin and Related Expansions

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19.5.5 $$q=\exp \left(-\pi K^{\prime }\left(k\right)/K\left(k\right)\right)=r+8r^2+84r^3+992r^4+\mathrm{\cdots },$$ $r=\frac{1}{16}{k}^2$, $0\le k\le 1$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K^{\prime }\left(k\right)$: Legendre’s complementary complete elliptic integral of the first kind, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\exp z$: exponential function, $q$: nome and $k$: real or complex modulus

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.6 $$q=\lambda +2{\lambda }^5+15{\lambda }^9+150{\lambda }^{13}+1707{\lambda }^{17}+\mathrm{\cdots },$$ $0\le k\le 1$,

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Symbols:

$q$: nome and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.8 $$K\left(k\right)=\frac{\pi }{2}{\left(1+2\sum _{n=1}^{\mathrm{\infty }}{q}^{n^2}\right)}^2,$$ ,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $q$: nome, $n$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5, §19.5

Permalink:

http://dlmf.nist.gov/19.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.10 $$K\left(k\right)=\frac{\pi }{2}\prod _{m=1}^{\mathrm{\infty }}(1+k_m),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $m$: nonnegative integer and $k$: real or complex modulus

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.5.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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19.5.11 $$k_{m+1}=\frac{1-\sqrt{1-k_m^2}}{1+\sqrt{1-k_m^2}},$$ $m=0,1,\mathrm{\dots }$.

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Symbols:

$m$: nonnegative integer and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.5.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §19.5 and Ch.19

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6: 22.17 Moduli Outside the Interval [0,1]

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22.17.1 $$\mathrm{p}\mathrm{q}(z,k)=\mathrm{p}\mathrm{q}(z,-k),$$

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Symbols:

$\mathrm{p}\mathrm{q}(z,k)$: generic Jacobian elliptic function, $z$: complex and $k$: modulus

Permalink:

http://dlmf.nist.gov/22.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §22.17(i), §22.17 and Ch.22

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22.17.6 $$\mathrm{sn}(z,\mathrm{i}k)=k_1^{\prime }\mathrm{sd}(z/k_1^{\prime },k_1),$$

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Symbols:

$\mathrm{sd}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\mathrm{i}$: imaginary unit, $z$: complex, $k$: modulus, $k_1$: change of variable and $k_1^{\prime }$: change of variable

A&S Ref:

16.10.2 (modified)

Permalink:

http://dlmf.nist.gov/22.17.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §22.17(i), §22.17 and Ch.22

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22.17.7 $$\mathrm{cn}(z,\mathrm{i}k)=\mathrm{cd}(z/k_1^{\prime },k_1),$$

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Symbols:

$\mathrm{cd}(z,k)$: Jacobian elliptic function, $\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{i}$: imaginary unit, $z$: complex, $k$: modulus, $k_1$: change of variable and $k_1^{\prime }$: change of variable

A&S Ref:

16.10.3 (modified)

Permalink:

http://dlmf.nist.gov/22.17.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §22.17(i), §22.17 and Ch.22

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22.17.8 $$\mathrm{dn}(z,\mathrm{i}k)=\mathrm{nd}(z/k_1^{\prime },k_1).$$

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Symbols:

$\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{nd}(z,k)$: Jacobian elliptic function, $\mathrm{i}$: imaginary unit, $z$: complex, $k$: modulus, $k_1$: change of variable and $k_1^{\prime }$: change of variable

A&S Ref:

16.10.4 (modified)

Permalink:

http://dlmf.nist.gov/22.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §22.17(i), §22.17 and Ch.22

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§22.17(ii) Complex Moduli

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7: 19.8 Quadratic Transformations

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19.8.5 $$K\left(k\right)=\frac{\pi }{2M(1,k^{\prime })},$$ .

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $k$: real or complex modulus and $k^{\prime }$: complementary modulus

Permalink:

http://dlmf.nist.gov/19.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.6 $$E\left(k\right)=\frac{\pi }{2M(1,k^{\prime })}\left(a_0^2-\sum _{n=0}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right)=K\left(k\right)\left(a_1^2-\sum _{n=2}^{\mathrm{\infty }}{2}^{n-1}{c}_n^2\right),$$ , $a_0=1$, $g_0=k^{\prime }$,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $E\left(k\right)$: Legendre’s complete elliptic integral of the second kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, $a_n$: iterate and $g_n$: iterate

Permalink:

http://dlmf.nist.gov/19.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.7 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{4M(1,k^{\prime })}\left(2+\frac{{\alpha }^2}{1-{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n\right),$$ , ,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter and $Q_n$

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.9 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{\pi }{4M(1,k^{\prime })}\frac{k^2}{k^2-{\alpha }^2}\sum _{n=0}^{\mathrm{\infty }}{Q}_n,$$ , ,

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Symbols:

$M(a,g)$: arithmetic-geometric mean, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $n$: nonnegative integer, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter and $Q_n$

Referenced by:

§19.8(i)

Permalink:

http://dlmf.nist.gov/19.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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19.8.10 $$p_0^2=1-(k^2/{\alpha }^2).$$

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Symbols:

$k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.8.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(i), §19.8 and Ch.19

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8: 19.25 Relations to Other Functions

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19.25.13 $$D(\varphi ,k)=\frac{1}{3}{R}_D(c-1,c-k^2,c).$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.25.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.17 $$F(\varphi ,k)=R_F(x,y,z),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.15

Permalink:

http://dlmf.nist.gov/19.25.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.18 $$(x,y,z)=(c-1,c-k^2,c),$$

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Symbols:

$k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.25.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.20 $$\mathrm{el1}(x,k_c)=R_F(r,r+k_c^2,r+1),$$

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Symbols:

$\mathrm{el1}(x,k_c)$: Bulirsch’s incomplete elliptic integral of the first kind, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $k$: real or complex modulus and $r$: inverse

Permalink:

http://dlmf.nist.gov/19.25.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(ii), §19.25 and Ch.19

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19.25.24 $${(z-x)}^{1/2}{R}_F(x,y,z)=F(\varphi ,k),$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.14(i), §19.25(iii)

Permalink:

http://dlmf.nist.gov/19.25.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(iii), §19.25 and Ch.19

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9: 19.6 Special Cases

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19.6.3 $$\mathrm{\Pi }({\alpha }^2,0)=\pi /(2\sqrt{1-{\alpha }^2}),\mathrm{\Pi }(0,k)=K\left(k\right),$$ .

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Permalink:

http://dlmf.nist.gov/19.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(i), §19.6 and Ch.19

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19.6.5 $$\mathrm{\Pi }({\alpha }^2,k)=K\left(k\right)-\mathrm{\Pi }(k^2/{\alpha }^2,k),$$

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Symbols:

$K\left(k\right)$: Legendre’s complete elliptic integral of the first kind, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.12, Figure 19.3.6, Figure 19.3.6, Figure 19.3.6, §19.6(i), §19.7(iii), §19.8(i)

Permalink:

http://dlmf.nist.gov/19.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(i), §19.6 and Ch.19

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19.6.10 $$\underset{\varphi \to 0}{lim}E(\varphi ,k)/\varphi =1.$$

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Symbols:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Permalink:

http://dlmf.nist.gov/19.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §19.6(iii), §19.6 and Ch.19

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10: 19.3 Graphics

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Figure 19.3.3: $F(\varphi ,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 2$, $0\le {\sin }^2\varphi \le 1$. …If ${\sin }^2\varphi =1/k^2$ (), then it has the value $K\left(1/k\right)/k$: put $c=k^2$ in (19.25.5) and use (19.25.1). Magnify 3D Help

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Figure 19.3.6: $\mathrm{\Pi }(\varphi ,2,k)$ as a function of $k^2$ and ${\sin }^2\varphi$ for $-1\le k^2\le 3$, . …If ${\sin }^2\varphi =1/k^2$ (), then by (19.7.4) it reduces to $\mathrm{\Pi }(2/k^2,1/k)/k$, $k^2\ne 2$, with Cauchy principal value $(K\left(1/k\right)-\mathrm{\Pi }(\frac{1}{2},1/k))/k$, , by (19.6.5). … Magnify 3D Help

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§19.3(ii) Complex Variables

►In Figures 19.3.7 and 19.3.8 for complete Legendre’s elliptic integrals with complex arguments, height corresponds to the absolute value of the function and color to the phase. … ►

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Figure 19.3.11: $\mathrm{\Re }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. …On the branch cut ($k^2>1$) it has the value $kE\left(1/k\right)+(k_{}^{\prime }{}_{}{}^2/k)K\left(1/k\right)$, with limit 1 as $k^2\to 1+$. Magnify 3D Help

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