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Inverse tangent

Table of contents: Definitions - Illustrations - Transcendental equations - Differential equations - Integral representations - Specific values - Analytic properties - Cases for atan2 - Argument transformations - Sums and products - Representations through other functions - Complex parts - Derivatives and integrals - Series expansions - Bounds and inequalities

Definitions

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Symbol: Atan — atan⁡(z)\operatorname{atan}(z) — Inverse tangent
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Symbol: Atan2 — atan2⁡ ⁣(y,x)\operatorname{atan2}\!\left(y, x\right) — Two-argument inverse tangent

Illustrations

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Image: X-ray of atan⁡(z)\operatorname{atan}(z) on z∈[−2,2]+[−2,2]iz \in \left[-2, 2\right] + \left[-2, 2\right] i

Transcendental equations

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tan⁡ ⁣(atan⁡(z))=z\tan\!\left(\operatorname{atan}(z)\right) = z
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sin⁡ ⁣(atan⁡(z))=z1+z2\sin\!\left(\operatorname{atan}(z)\right) = \frac{z}{\sqrt{1 + {z}^{2}}}
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cos⁡ ⁣(atan⁡(z))=11+z2\cos\!\left(\operatorname{atan}(z)\right) = \frac{1}{\sqrt{1 + {z}^{2}}}
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atan⁡ ⁣(tan⁡(θ))=θ\operatorname{atan}\!\left(\tan(\theta)\right) = \theta
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solutions⁡ w∈C[tan⁡(w)=z]={atan⁡(z)+πn:n∈Z}\mathop{\operatorname{solutions}\,}\limits_{w \in \mathbb{C}} \left[\tan(w) = z\right] = \left\{ \operatorname{atan}(z) + \pi n : n \in \mathbb{Z} \right\}
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atan2⁡ ⁣(y,x)=solution*⁡ θ∈(−π,π][(x,y)=(rcos⁡(θ),rsin⁡(θ))   where r=x2+y2]\operatorname{atan2}\!\left(y, x\right) = \mathop{\operatorname{solution*}\,}\limits_{\theta \in \left(-\pi, \pi\right]} \left[\left(x, y\right) = \left(r \cos(\theta), r \sin(\theta)\right)\; \text{ where } r = \sqrt{{x}^{2} + {y}^{2}}\right]

Differential equations

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(1+z2)y′′(z)+2zy′(z)=0   where y(z)=c1+c2atan⁡(z)\left(1 + {z}^{2}\right) y''(z) + 2 z y'(z) = 0\; \text{ where } y(z) = {c}_{1} + {c}_{2} \operatorname{atan}(z)

Integral representations

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atan⁡(z)=∫0z11+t2 dt\operatorname{atan}(z) = \int_{0}^{z} \frac{1}{1 + {t}^{2}} \, dt

Specific values

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atan⁡(0)=0\operatorname{atan}(0) = 0
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atan⁡ ⁣(+∞)=π2\operatorname{atan}\!\left(+\infty\right) = \frac{\pi}{2}
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atan⁡ ⁣(−∞)=−π2\operatorname{atan}\!\left(-\infty\right) = -\frac{\pi}{2}
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atan⁡ ⁣(+i)=+i∞\operatorname{atan}\!\left(+i\right) = +i \infty
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atan⁡ ⁣(−i)=−i∞\operatorname{atan}\!\left(-i\right) = -i \infty
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atan⁡(1)=π4\operatorname{atan}(1) = \frac{\pi}{4}
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atan⁡ ⁣(3)=π3\operatorname{atan}\!\left(\sqrt{3}\right) = \frac{\pi}{3}
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atan⁡ ⁣(13)=π6\operatorname{atan}\!\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}
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atan⁡ ⁣(2−1)=π8\operatorname{atan}\!\left(\sqrt{2} - 1\right) = \frac{\pi}{8}
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atan⁡ ⁣(2+1)=3π8\operatorname{atan}\!\left(\sqrt{2} + 1\right) = \frac{3 \pi}{8}
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atan⁡ ⁣(2−3)=π12\operatorname{atan}\!\left(2 - \sqrt{3}\right) = \frac{\pi}{12}
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atan⁡ ⁣(2+3)=5π12\operatorname{atan}\!\left(2 + \sqrt{3}\right) = \frac{5 \pi}{12}

Analytic properties

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atan⁡(z) is holomorphic on z∈C∖((−∞,−1]i∪[1,∞)i)\operatorname{atan}(z) \text{ is holomorphic on } z \in \mathbb{C} \setminus \left(\left(-\infty, -1\right] i \cup \left[1, \infty\right) i\right)
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EssentialSingularities⁡ ⁣(atan⁡(z),z,C∪{∞~})={}\operatorname{EssentialSingularities}\!\left(\operatorname{atan}(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{\right\}
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poles⁡ z∈C∪{∞~}atan⁡(z)={}\mathop{\operatorname{poles}\,}\limits_{z \in \mathbb{C} \cup \left\{{\tilde \infty}\right\}} \operatorname{atan}(z) = \left\{\right\}
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BranchPoints⁡ ⁣(atan⁡(z),z,C∪{∞~})={−i,i,∞~}\operatorname{BranchPoints}\!\left(\operatorname{atan}(z), z, \mathbb{C} \cup \left\{{\tilde \infty}\right\}\right) = \left\{-i, i, {\tilde \infty}\right\}
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BranchCuts⁡ ⁣(atan⁡(z),z,C)={(−∞,−1]i,[1,∞)i}\operatorname{BranchCuts}\!\left(\operatorname{atan}(z), z, \mathbb{C}\right) = \left\{\left(-\infty, -1\right] i, \left[1, \infty\right) i\right\}
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zeros⁡ z∈Catan⁡(z)={0}\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \operatorname{atan}(z) = \left\{0\right\}

Cases for atan2

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atan2⁡ ⁣(0,x)={0,x≥0π,x<0\operatorname{atan2}\!\left(0, x\right) = \begin{cases} 0, & x \ge 0\\\pi, & x < 0\\ \end{cases}
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atan2⁡ ⁣(y,0)=π2sgn⁡(y)\operatorname{atan2}\!\left(y, 0\right) = \frac{\pi}{2} \operatorname{sgn}(y)
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atan2⁡ ⁣(y,x)={0,x=y=0atan⁡ ⁣(yx),x>0(π2)sgn⁡(y)−atan⁡ ⁣(xy),y≠0π,y=0  and⁡  x<0\operatorname{atan2}\!\left(y, x\right) = \begin{cases} 0, & x = y = 0\\\operatorname{atan}\!\left(\frac{y}{x}\right), & x > 0\\\left(\frac{\pi}{2}\right) \operatorname{sgn}(y) - \operatorname{atan}\!\left(\frac{x}{y}\right), & y \ne 0\\\pi, & y = 0 \;\mathbin{\operatorname{and}}\; x < 0\\ \end{cases}

Argument transformations

Symmetries

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atan⁡ ⁣(−z)=−atan⁡(z)\operatorname{atan}\!\left(-z\right) = -\operatorname{atan}(z)
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atan⁡ ⁣(z‾)=atan⁡(z)‾\operatorname{atan}\!\left(\overline{z}\right) = \overline{\operatorname{atan}(z)}
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atan⁡ ⁣(1z)=π2−atan⁡(z)\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} - \operatorname{atan}(z)
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atan⁡ ⁣(1z)=−π2−atan⁡(z)\operatorname{atan}\!\left(\frac{1}{z}\right) = -\frac{\pi}{2} - \operatorname{atan}(z)
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atan⁡ ⁣(1z)=π2csgn⁡ ⁣(1z)−atan⁡(z)\operatorname{atan}\!\left(\frac{1}{z}\right) = \frac{\pi}{2} \operatorname{csgn}\!\left(\frac{1}{z}\right) - \operatorname{atan}(z)

Addition and multiplication formulas

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atan⁡ ⁣(iz)=iatanh⁡(z)\operatorname{atan}\!\left(i z\right) = i \operatorname{atanh}(z)
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atan⁡ ⁣(x+y)=atan⁡(x)+atan⁡ ⁣(y1+x(x+y))\operatorname{atan}\!\left(x + y\right) = \operatorname{atan}(x) + \operatorname{atan}\!\left(\frac{y}{1 + x \left(x + y\right)}\right)
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atan⁡ ⁣(2z)=atan⁡(z)+atan⁡ ⁣(z1+2z2)\operatorname{atan}\!\left(2 z\right) = \operatorname{atan}(z) + \operatorname{atan}\!\left(\frac{z}{1 + 2 {z}^{2}}\right)

Algebraic transformations

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atan⁡(z)=2atan⁡ ⁣(z1+1+z2)\operatorname{atan}(z) = 2 \operatorname{atan}\!\left(\frac{z}{1 + \sqrt{1 + {z}^{2}}}\right)

Sums and products

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atan⁡(x)+atan⁡(y)=atan2⁡ ⁣(x+y,1−xy)\operatorname{atan}(x) + \operatorname{atan}(y) = \operatorname{atan2}\!\left(x + y, 1 - x y\right)
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atan⁡(x)−atan⁡(y)=atan2⁡ ⁣(x−y,1+xy)\operatorname{atan}(x) - \operatorname{atan}(y) = \operatorname{atan2}\!\left(x - y, 1 + x y\right)
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atan⁡(x)+atan⁡(y)=atan⁡ ⁣(x+y1−xy)\operatorname{atan}(x) + \operatorname{atan}(y) = \operatorname{atan}\!\left(\frac{x + y}{1 - x y}\right)
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atan⁡(x)−atan⁡(y)=atan⁡ ⁣(x−y1+xy)\operatorname{atan}(x) - \operatorname{atan}(y) = \operatorname{atan}\!\left(\frac{x - y}{1 + x y}\right)
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atan2⁡ ⁣(y1,x1)+atan2⁡ ⁣(y2,x2)=atan2⁡ ⁣(y1x2+y2x1,x1x2−y1y2)\operatorname{atan2}\!\left({y}_{1}, {x}_{1}\right) + \operatorname{atan2}\!\left({y}_{2}, {x}_{2}\right) = \operatorname{atan2}\!\left({y}_{1} {x}_{2} + {y}_{2} {x}_{1}, {x}_{1} {x}_{2} - {y}_{1} {y}_{2}\right)
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atan2⁡ ⁣(y1,x1)−atan2⁡ ⁣(y2,x2)=atan2⁡ ⁣(y1x2−y2x1,x1x2+y1y2)\operatorname{atan2}\!\left({y}_{1}, {x}_{1}\right) - \operatorname{atan2}\!\left({y}_{2}, {x}_{2}\right) = \operatorname{atan2}\!\left({y}_{1} {x}_{2} - {y}_{2} {x}_{1}, {x}_{1} {x}_{2} + {y}_{1} {y}_{2}\right)

Representations through other functions

Logarithms

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atan⁡(z)=i2(log⁡ ⁣(1−iz)−log⁡ ⁣(1+iz))\operatorname{atan}(z) = \frac{i}{2} \left(\log\!\left(1 - i z\right) - \log\!\left(1 + i z\right)\right)
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atan⁡(z)=i2log⁡ ⁣(1−iz1+iz)\operatorname{atan}(z) = \frac{i}{2} \log\!\left(\frac{1 - i z}{1 + i z}\right)
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atan⁡(z)=−i2log⁡ ⁣(1+iz1−iz)\operatorname{atan}(z) = -\frac{i}{2} \log\!\left(\frac{1 + i z}{1 - i z}\right)
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atan2⁡ ⁣(y,x)=−ilog⁡ ⁣(sgn⁡ ⁣(x+yi))\operatorname{atan2}\!\left(y, x\right) = -i \log\!\left(\operatorname{sgn}\!\left(x + y i\right)\right)
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atan2⁡ ⁣(y,x)=Im⁡ ⁣(log⁡ ⁣(x+yi))\operatorname{atan2}\!\left(y, x\right) = \operatorname{Im}\!\left(\log\!\left(x + y i\right)\right)

Inverse trigonometric functions

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atan⁡(z)=acot⁡ ⁣(1z)\operatorname{atan}(z) = \operatorname{acot}\!\left(\frac{1}{z}\right)
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atan⁡(z)=asin⁡ ⁣(z1+z2)\operatorname{atan}(z) = \operatorname{asin}\!\left(\frac{z}{\sqrt{1 + {z}^{2}}}\right)
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atan⁡(z)=csgn⁡(z)acos⁡ ⁣(11+z2)\operatorname{atan}(z) = \operatorname{csgn}(z) \operatorname{acos}\!\left(\frac{1}{\sqrt{1 + {z}^{2}}}\right)

Hypergeometric functions

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atan⁡(z)=z 2F1 ⁣(1,12,32,−z2)\operatorname{atan}(z) = z \,{}_2F_1\!\left(1, \frac{1}{2}, \frac{3}{2}, -{z}^{2}\right)

Complex parts

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Re⁡ ⁣(atan⁡ ⁣(x+yi))=12atan2⁡ ⁣(2x,1−x2−y2)\operatorname{Re}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{2} \operatorname{atan2}\!\left(2 x, 1 - {x}^{2} - {y}^{2}\right)
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Im⁡ ⁣(atan⁡ ⁣(x+yi))=14log⁡ ⁣(x2+(1+y)2x2+(1−y)2)\operatorname{Im}\!\left(\operatorname{atan}\!\left(x + y i\right)\right) = \frac{1}{4} \log\!\left(\frac{{x}^{2} + {\left(1 + y\right)}^{2}}{{x}^{2} + {\left(1 - y\right)}^{2}}\right)

Derivatives and integrals

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atan⁡′(z)=11+z2\operatorname{atan}'(z) = \frac{1}{1 + {z}^{2}}
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atan⁡′′(z)=−2z(1+z2)2\operatorname{atan}''(z) = -\frac{2 z}{{\left(1 + {z}^{2}\right)}^{2}}
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atan⁡(n)(z)=(n−1)!(1+z2)(n+1)/2Un−1 ⁣(−z1+z2){\operatorname{atan}}^{(n)}(z) = \frac{\left(n - 1\right)!}{{\left(1 + {z}^{2}\right)}^{\left( n + 1 \right) / 2}} U_{n - 1}\!\left(-\frac{z}{\sqrt{1 + {z}^{2}}}\right)
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atan⁡(n)(z)=(−1)n(n−1)!2i(1(z+i)n−1(z−i)n){\operatorname{atan}}^{(n)}(z) = \frac{{\left(-1\right)}^{n} \left(n - 1\right)!}{2 i} \left(\frac{1}{{\left(z + i\right)}^{n}} - \frac{1}{{\left(z - i\right)}^{n}}\right)
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ddx atan2⁡ ⁣(y,x)=−yx2+y2\frac{d}{d x}\, \operatorname{atan2}\!\left(y, x\right) = -\frac{y}{{x}^{2} + {y}^{2}}
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ddy atan2⁡ ⁣(y,x)=xx2+y2\frac{d}{d y}\, \operatorname{atan2}\!\left(y, x\right) = \frac{x}{{x}^{2} + {y}^{2}}

Series expansions

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atan⁡(z)=∑k=0∞(−1)kz2k+12k+1\operatorname{atan}(z) = \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k} {z}^{2 k + 1}}{2 k + 1}

Bounds and inequalities

Real arguments

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∣atan2⁡ ⁣(y,x)∣≤π\left|\operatorname{atan2}\!\left(y, x\right)\right| \le \pi
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∣atan⁡(x)∣<π2\left|\operatorname{atan}(x)\right| < \frac{\pi}{2}
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∣atan⁡(x)∣≤π2\left|\operatorname{atan}(x)\right| \le \frac{\pi}{2}
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∣atan⁡(x)∣≤∣x∣\left|\operatorname{atan}(x)\right| \le \left|x\right|
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atan⁡(x)≤∑k=02N(−1)kx2k+12k+1\operatorname{atan}(x) \le \sum_{k=0}^{2 N} \frac{{\left(-1\right)}^{k} {x}^{2 k + 1}}{2 k + 1}
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atan⁡(x)≥∑k=02N+1(−1)kx2k+12k+1\operatorname{atan}(x) \ge \sum_{k=0}^{2 N + 1} \frac{{\left(-1\right)}^{k} {x}^{2 k + 1}}{2 k + 1}
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atan⁡(x)≤(π2)2x1+π2x\operatorname{atan}(x) \le {\left(\frac{\pi}{2}\right)}^{2} \frac{x}{1 + \frac{\pi}{2} x}
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atan⁡(x)≥x1+x\operatorname{atan}(x) \ge \frac{x}{1 + x}
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atan⁡(x)≥πxπ+2x\operatorname{atan}(x) \ge \frac{\pi x}{\pi + 2 x}
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atan⁡(x)≤π2x1+x2\operatorname{atan}(x) \le \frac{\pi}{2} \frac{x}{\sqrt{1 + {x}^{2}}}
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atan⁡(x)≥x1+x2\operatorname{atan}(x) \ge \frac{x}{\sqrt{1 + {x}^{2}}}
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atan⁡(x)≤π2tanh⁡(x)\operatorname{atan}(x) \le \frac{\pi}{2} \tanh(x)
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atan⁡(x)≥tanh⁡(x)\operatorname{atan}(x) \ge \tanh(x)

Complex arguments

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∣atan⁡(z)∣≤∣atanh⁡ ⁣(∣z∣)∣\left|\operatorname{atan}(z)\right| \le \left|\operatorname{atanh}\!\left(\left|z\right|\right)\right|
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∣atan⁡(z)∣≤−log⁡ ⁣(1−∣z∣)\left|\operatorname{atan}(z)\right| \le -\log\!\left(1 - \left|z\right|\right)

Perturbations

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∣atan⁡ ⁣(x+y)−atan⁡(x)∣=atan2⁡ ⁣(∣y∣,1+x(x+y))\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}(x)\right| = \operatorname{atan2}\!\left(\left|y\right|, 1 + x \left(x + y\right)\right)
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∣atan⁡ ⁣(x+y)−atan⁡(x)∣≤∣y∣\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}(x)\right| \le \left|y\right|
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∣atan⁡ ⁣(x+y)−atan⁡(x)∣≤∣y∣1+(max⁡ ⁣(0,∣x∣−∣y∣))2\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}(x)\right| \le \frac{\left|y\right|}{1 + {\left(\max\!\left(0, \left|x\right| - \left|y\right|\right)\right)}^{2}}
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∣atan⁡ ⁣(x+y)−atan⁡(x)∣<π\left|\operatorname{atan}\!\left(x + y\right) - \operatorname{atan}(x)\right| < \pi

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC