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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}\!\left(z\right) Sign function
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Symbol: Abs z\left|z\right| Absolute value
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Symbol: Arg arg ⁣(z)\arg\!\left(z\right) Complex argument
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Symbol: Re Re ⁣(z)\operatorname{Re}\!\left(z\right) Real part
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Symbol: Im Im ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
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Symbol: Conjugate z\overline{z} Complex conjugate

Basic formulas

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sgn ⁣(z)=zz\operatorname{sgn}\!\left(z\right) = \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

Specific values

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arg ⁣(1)=0\arg\!\left(1\right) = 0
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arg ⁣(i)=π2\arg\!\left(i\right) = \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\!\left(-1\right) = \pi

Connection formulas

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

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}\!\left(z\right)\right| \le \left|z\right|
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Im ⁣(z)z\left|\operatorname{Im}\!\left(z\right)\right| \le \left|z\right|
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zRe ⁣(z)+Im ⁣(z)\left|z\right| \le \left|\operatorname{Re}\!\left(z\right)\right| + \left|\operatorname{Im}\!\left(z\right)\right|
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sgn ⁣(z)1\left|\operatorname{sgn}\!\left(z\right)\right| \le 1
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arg ⁣(z)π\left|\arg\!\left(z\right)\right| \le \pi

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

2019-05-23 08:00:13.607731 UTC