• Thumbnail for Imaginary unit
    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...
    30 KB (4,171 words) - 14:39, 16 May 2025
  • Thumbnail for Complex number
    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...
    92 KB (12,059 words) - 18:44, 26 May 2025
  • 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...
    12 KB (1,347 words) - 02:23, 8 May 2025
  • .. (sequence A001113 in the OEIS). The imaginary unit or unit imaginary number, denoted as i, is a mathematical concept which extends...
    40 KB (3,556 words) - 19:55, 28 May 2025
  • Thumbnail for Error function
    {\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...
    47 KB (7,328 words) - 08:39, 27 April 2025
  • 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...
    9 KB (1,154 words) - 22:24, 25 March 2025
  • Thumbnail for Euler's formula
    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...
    26 KB (3,851 words) - 04:01, 16 April 2025
  • Thumbnail for Euler's identity
    number, the base of natural logarithms, i {\displaystyle i} is the imaginary unit, which by definition satisfies i 2 = − 1 {\displaystyle i^{2}=-1} ,...
    15 KB (1,947 words) - 13:10, 10 April 2025
  • Versor (redirect from Unit quaternion)
    } 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...
    22 KB (2,945 words) - 20:20, 24 May 2025
  • 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...
    11 KB (1,489 words) - 11:08, 5 February 2025
  • 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...
    7 KB (906 words) - 01:57, 5 September 2024
  • Thumbnail for Hyperbolic functions
    Roger Cotes suggested to modify the trigonometric functions using the imaginary unit i = − 1 {\displaystyle i={\sqrt {-1}}} to obtain an oblate spheroid...
    31 KB (4,989 words) - 21:48, 30 April 2025
  • 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....
    21 KB (3,162 words) - 21:52, 18 April 2025
  • 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...
    5 KB (689 words) - 13:57, 25 May 2025
  • 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...
    17 KB (1,922 words) - 14:43, 16 May 2025
  • Thumbnail for Leonhard Euler
    value of a function, the letter i {\displaystyle i} to express the imaginary unit − 1 {\displaystyle {\sqrt {-1}}} , the Greek letter Σ {\displaystyle...
    107 KB (10,831 words) - 13:51, 2 May 2025
  • Thumbnail for J
    J
    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...
    26 KB (2,024 words) - 20:33, 25 May 2025
  • {\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...
    17 KB (3,260 words) - 21:19, 22 May 2025
  • real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit i , {\displaystyle i,} then it becomes Hermitian. Every Hermitian matrix...
    20 KB (3,028 words) - 01:11, 26 May 2025
  • -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...
    3 KB (588 words) - 18:53, 24 February 2025
  • Thumbnail for Number
    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...
    66 KB (8,359 words) - 04:26, 12 May 2025
  • 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,...
    21 KB (3,019 words) - 12:55, 23 January 2025
  • 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...
    9 KB (1,163 words) - 19:24, 15 December 2024
  • 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...
    28 KB (4,144 words) - 21:21, 22 March 2025
  • 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...
    6 KB (812 words) - 11:27, 30 January 2025
  • {\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 ∂...
    15 KB (2,146 words) - 17:26, 28 May 2025
  • Thumbnail for Group of rational points on the unit circle
    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...
    8 KB (1,226 words) - 07:08, 10 May 2024
  • Thumbnail for Fresnel equations
    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)...
    68 KB (8,252 words) - 04:50, 10 March 2025
  • 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...
    32 KB (4,773 words) - 18:22, 4 April 2025
  • 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...
    25 KB (2,352 words) - 17:24, 7 February 2025