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...

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...

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

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...

{\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...

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...

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....

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

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...

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...

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

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...

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...

}({\hat {\mathbf {k} }}\cdot {\hat {\mathbf {r} }}),} where i is the imaginary unit, k is a wave vector of length k, r is a position vector of length r...

and imaginary (y) parts: z = x + i y {\displaystyle z=x+iy} for example: z = 4 + 5i, where x and y are real numbers, and i is the imaginary unit. In this...

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...

bivector part. The coordinates w, x, y, z are complex numbers with imaginary unit h: x = x 1 + h x 2 ,   y = y 1 + h y 2 ,   z = z 1 + h z 2 , h 2 = −...

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...

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...

where the square root is i = − 1 , {\displaystyle i={\sqrt {-1}},} the imaginary unit. As ( a + b d ) ( a − b d ) = a 2 − b 2 d {\displaystyle (a+b{\sqrt...

{\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 ∂...

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

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

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...

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...

{\displaystyle e} is Euler's number, and i {\displaystyle i} is the imaginary unit. Functions of this form are known as Bloch functions or Bloch states...

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...

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,...