The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x2 + 1 = 0. Although there is no...

extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation i 2 = − 1 {\displaystyle i^{2}=-1} ; every...

An imaginary number is the product of a real number and the imaginary unit i, which is defined by its property i2 = −1. The square of an imaginary number...

.. (sequence A001113 in the OEIS). The imaginary unit or unit imaginary number, denoted as i, is a mathematical concept which extends...

{\displaystyle \operatorname {erfi} z=-i\operatorname {erf} iz,} where i is the imaginary unit. The name "error function" and its abbreviation erf were proposed by...

Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit i. Imaginary time is...

x+i\sin x,} where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively...

number, the base of natural logarithms, i {\displaystyle i} is the imaginary unit, which by definition satisfies i 2 = − 1 {\displaystyle i^{2}=-1} ,...

} where the r2 = −1 condition means that r is an imaginary unit. There is a sphere of imaginary units in the quaternions. Note that the expression for...

mechanics formally akin to imaginary time in quantum mechanics: that is, it, where t is time and i is the imaginary unit (i2 = –1). More precisely, in...

conductance (siemens); B is the susceptance (siemens); and j2 = −1, the imaginary unit. The dynamic effects of the material's susceptance relate to the universal...

Roger Cotes suggested to modify the trigonometric functions using the imaginary unit i = − 1 {\displaystyle i={\sqrt {-1}}} to obtain an oblate spheroid...

In Hamilton quaternions there is a sphere of imaginary units, and any pair of antipodal imaginary units generates a complex plane with the real line....

which satisfies j 2 = − 1. {\displaystyle j^{2}=-1.} Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a...

vector part v is a unit vector in R 3 {\displaystyle \mathbb {R} ^{3}} . Thus the right versors extend the notion of imaginary units found in the complex...

value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle...

the SI derived unit for energy. In some areas of physics, electrical engineering and related fields, j is the symbol for the imaginary unit (the square root...

{\big (}\cos x+i\sin x{\big )}^{n}=\cos nx+i\sin nx,} where i is the imaginary unit (i2 = −1). The formula is named after Abraham de Moivre, although he...

real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit i , {\displaystyle i,} then it becomes Hermitian. Every Hermitian matrix...

-th Bessel function of the first kind and i {\displaystyle i} is the imaginary unit, i 2 = − 1. {\textstyle i^{2}=-1.} Substituting θ {\textstyle \theta...

The hypercomplex numbers include one real unit together with 2 n − 1 {\displaystyle 2^{n}-1} imaginary units, for which n is a non-negative integer. For...

of x and px , i is the imaginary unit, and ℏ is the reduced Planck constant h/2π, and I {\displaystyle \mathbb {I} } is the unit operator. In general,...

impedance (Z = R + jX), where the imaginary part is reactance (X) and the real part is resistance (R). In SI units, susceptance is measured in siemens...

numbers and the hyperbolic unit j satisfies j 2 = + 1 {\displaystyle j^{2}=+1} In the field of complex numbers the imaginary unit i satisfies i 2 = − 1. {\displaystyle...

symbol used to denote the Bessel function j, used as the symbol for the imaginary unit ( − 1 {\displaystyle {\sqrt {-1}}} ) in fields where i is used for a...

{\partial }{\partial x}}} where ħ is the reduced Planck constant, i the imaginary unit, x is the spatial coordinate, and a partial derivative (denoted by ∂...

on the unit circle there exists the rational point (a/c, b/c), which, in the complex plane, is just a/c + ib/c, where i is the imaginary unit. Conversely...

form where Ek is the (constant) complex amplitude vector, i is the imaginary unit, k is the wave vector (whose magnitude k is the angular wavenumber)...

has two imaginary square roots, ⁠ + i x {\displaystyle +i{\sqrt {x}}} ⁠ and ⁠ − i x {\displaystyle -i{\sqrt {x}}} ⁠, where i is the imaginary unit. In general...

cis x = cos x + i sin x, where cos is the cosine function, i is the imaginary unit and sin is the sine function. x is the argument of the complex number...