Imaginary part

Imaginary part

In mathematics, the imaginary part of a complex number z, is the second element of the ordered pair of real numbers representing z, i.e. if z = (x, y) , or equivalently, z = x+iy, then the imaginary part of z is y. It is denoted by Im (z) or Im{"z"}, where Im is a capital I in the Fraktur typeface. The complex function which maps z to the imaginary part of z is not holomorphic.

In terms of the complex conjugate ar{z}, the imaginary part of "z" is equal to frac{z-ar{z{2i}.

For a complex number in polar form, z = (r, heta ), or equivalently, z = r(cos heta + i sin heta) , it follows from Euler's formula that z = re^{i heta}, and hence that the imaginary part of re^{i heta} is rsin heta.

In electric power, when a sine wave voltage drives a "linear" load (in other words, a load that makes the current also be a sine wave), the current I in the power wires can be represented as a complex number I = x + jy (engineers use j to indicate the imaginary unit rather than i, which also represents current). The "real current" x is related to the current when the voltage is maximum. The real current times the voltage gives the actual power consumed by the load (often all that power is dissipated as heat). The "imaginary current" y is related to the current when the voltage is zero. A load with purely imaginary current (such as a capacitor or inductor) dissipates no power; it merely accepts power temporarily then later pushes that power back on the power lines.

See also

*Real part
*Imaginary number


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • imaginary part — n. the coefficient of the square root of negative one in a complex number as 5 in ( 3 + 5i): formerly, this coefficient multiplied by i was considered the imaginary part …   English World dictionary

  • imaginary part — noun the part of a complex number that has the square root of 1 as a factor • Syn: ↑imaginary part of a complex number • Hypernyms: ↑pure imaginary number • Part Holonyms: ↑complex number, ↑complex quantity, ↑ …   Useful english dictionary

  • imaginary part — menamoji dalis statusas T sritis automatika atitikmenys: angl. imaginary part vok. Imaginärteil, m rus. мнимая часть, f pranc. partie imaginaire, f …   Automatikos terminų žodynas

  • imaginary part — menamoji dalis statusas T sritis fizika atitikmenys: angl. imaginary part vok. Imaginärteil, m rus. мнимая часть, f pranc. partie imaginaire, f …   Fizikos terminų žodynas

  • imaginary part of a complex number — noun the part of a complex number that has the square root of 1 as a factor • Syn: ↑imaginary part • Hypernyms: ↑pure imaginary number • Part Holonyms: ↑complex number, ↑complex quantity, ↑imaginary number, ↑ …   Useful english dictionary

  • imaginary part of complex permeability — menamoji magnetinė skvarba statusas T sritis fizika atitikmenys: angl. imaginary part of complex permeability; reactive permeability vok. Blindpermeabilität, f; Imaginärteil der komplexen Permeabilität, m; Reihenwiderstandspermeabilität, f rus.… …   Fizikos terminų žodynas

  • imaginary part — noun Date: 1845 the part of a complex number (as 3i in 2 + 3i) that has the imaginary unit as a factor …   New Collegiate Dictionary

  • imaginary part — Math. the coefficient b in the complex number a + bi. Cf. real part. [1925 30] * * * …   Universalium

  • imaginary part — noun Of a complex number a + bi, the value b …   Wiktionary

  • Imaginary number — In mathematics, an imaginary number (or purely imaginary number) is a complex number whose squared value is a real number not greater than zero.Ahlfors, Lars V. Complex Analysis. 3rd Ed. New York: McGraw Hill, 1979. Pages 1 4.] The imaginary unit …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”