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Complex parts

Table of contents: Basic formulas - Specific values - Connection formulas - Functional equations - Bounds and inequalities

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Symbol: Sign sgn(z)\operatorname{sgn}(z) Sign function
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Symbol: Abs z\left|z\right| Absolute value
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Symbol: Arg arg(z)\arg(z) Complex argument
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Symbol: Re Re(z)\operatorname{Re}(z) Real part
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Symbol: Im Im(z)\operatorname{Im}(z) Imaginary part
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Symbol: Conjugate z\overline{z} Complex conjugate
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Symbol: Csgn csgn(z)\operatorname{csgn}(z) Real-valued sign function for complex numbers

Basic formulas

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sgn(z)=zz\operatorname{sgn}(z) = \frac{z}{\left|z\right|}
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x+yi=x2+y2\left|x + y i\right| = \sqrt{{x}^{2} + {y}^{2}}
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arg ⁣(x+yi)=atan2 ⁣(y,x)\arg\!\left(x + y i\right) = \operatorname{atan2}\!\left(y, x\right)
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Re ⁣(x+yi)=x\operatorname{Re}\!\left(x + y i\right) = x
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Im ⁣(x+yi)=y\operatorname{Im}\!\left(x + y i\right) = y
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x+yi=xyi\overline{x + y i} = x - y i
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sgn(x)={1,x>01,x<00,x=0\operatorname{sgn}(x) = \begin{cases} 1, & x > 0\\-1, & x < 0\\0, & x = 0\\ \end{cases}
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csgn(z)={sgn ⁣(Im(z)),Re(z)=0sgn ⁣(Re(z)),otherwise\operatorname{csgn}(z) = \begin{cases} \operatorname{sgn}\!\left(\operatorname{Im}(z)\right), & \operatorname{Re}(z) = 0\\\operatorname{sgn}\!\left(\operatorname{Re}(z)\right), & \text{otherwise}\\ \end{cases}
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csgn(z)=z2z\operatorname{csgn}(z) = \frac{\sqrt{{z}^{2}}}{z}

Specific values

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arg(1)=0\arg(1) = 0
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arg(i)=π2\arg(i) = \frac{\pi}{2}
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arg ⁣(i)=π2\arg\!\left(-i\right) = -\frac{\pi}{2}
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arg(1)=π\arg(-1) = \pi

Connection formulas

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Re(z)=z+z2\operatorname{Re}(z) = \frac{z + \overline{z}}{2}
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Im(z)=zz2i\operatorname{Im}(z) = \frac{z - \overline{z}}{2 i}
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sgn(z)=exp ⁣(iarg(z))\operatorname{sgn}(z) = \exp\!\left(i \arg(z)\right)
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arg(z)=ilog ⁣(sgn(z))\arg(z) = -i \log\!\left(\operatorname{sgn}(z)\right)

Functional equations

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zz=z2z \overline{z} = {\left|z\right|}^{2}
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arg ⁣(cz)=arg(z)\arg\!\left(c z\right) = \arg(z)

Bounds and inequalities

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ab=ab\left|a b\right| = \left|a\right| \left|b\right|
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a+ba+b\left|a + b\right| \le \left|a\right| + \left|b\right|
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abab\left|\left|a\right| - \left|b\right|\right| \le \left|a - b\right|
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z=z\left|\overline{z}\right| = \left|z\right|
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Re(z)z\left|\operatorname{Re}(z)\right| \le \left|z\right|
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Im(z)z\left|\operatorname{Im}(z)\right| \le \left|z\right|
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zRe(z)+Im(z)\left|z\right| \le \left|\operatorname{Re}(z)\right| + \left|\operatorname{Im}(z)\right|
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sgn(z)1\left|\operatorname{sgn}(z)\right| \le 1
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arg(z)π\left|\arg(z)\right| \le \pi

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

2019-11-11 15:50:15.016492 UTC