# discriminant

(0.002 seconds)

## 8 matching pages

##### 1: 1.11 Zeros of Polynomials
The discriminant of $f(z)$ is defined by
1.11.9 $D=a_{n}^{2n-2}\prod_{j
The discriminant of $g(w)$ is
1.11.12 $D=-4p^{3}-27q^{2}.$
The discriminant of $g(w)$ is …
##### 2: 27.14 Unrestricted Partitions
###### §27.14(vi) Ramanujan’s Tau Function
The discriminant function $\Delta\left(\tau\right)$ is defined by
27.14.16 $\Delta\left(\tau\right)=(2\pi)^{12}(\eta\left(\tau\right))^{24},$ $\Im\tau>0$,
27.14.17 $\Delta\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{12}\Delta\left(\tau% \right),$
##### 4: 23.3 Differential Equations
###### §23.3(i) Invariants, Roots, and Discriminant
The discriminant1.11(ii)) is given by
##### 5: 28.29 Definitions and Basic Properties
###### §28.29(iii) Discriminant and Eigenvalues in the Real Case
28.29.15 $\bigtriangleup(\lambda)=w_{\mbox{\tiny I}}(\pi,\lambda)+w_{\mbox{\tiny II}}^{% \prime}(\pi,\lambda)$
is called the discriminant of (28.29.1). … For this purpose the discriminant can be expressed as an infinite determinant involving the Fourier coefficients of $Q(x)$; see Magnus and Winkler (1966, §2.3, pp. 28–36). …
##### 6: 23.1 Special Notation
 $\mathbb{L}$ lattice in $\mathbb{C}$. … discriminant ${g_{2}}^{3}-27{g_{3}}^{2}$. …
##### 7: 23.22 Methods of Computation
• (a)

In the general case, given by $cd\neq 0$, we compute the roots $\alpha$, $\beta$, $\gamma$, say, of the cubic equation $4t^{3}-ct-d=0$; see §1.11(iii). These roots are necessarily distinct and represent $e_{1}$, $e_{2}$, $e_{3}$ in some order.

If $c$ and $d$ are real, and the discriminant is positive, that is $c^{3}-27d^{2}>0$, then $e_{1}$, $e_{2}$, $e_{3}$ can be identified via (23.5.1), and $k^{2}$, ${k^{\prime}}^{2}$ obtained from (23.6.16).

If $c^{3}-27d^{2}<0$, or $c$ and $d$ are not both real, then we label $\alpha$, $\beta$, $\gamma$ so that the triangle with vertices $\alpha$, $\beta$, $\gamma$ is positively oriented and $[\alpha,\gamma]$ is its longest side (chosen arbitrarily if there is more than one). In particular, if $\alpha$, $\beta$, $\gamma$ are collinear, then we label them so that $\beta$ is on the line segment $(\alpha,\gamma)$. In consequence, $k^{2}=(\beta-\gamma)/(\alpha-\gamma)$, ${k^{\prime}}^{2}=(\alpha-\beta)/(\alpha-\gamma)$ satisfy $\Im k^{2}\geq 0\geq\Im{k^{\prime}}^{2}$ (with strict inequality unless $\alpha$, $\beta$, $\gamma$ are collinear); also $|k^{2}|$, $|{k^{\prime}}^{2}|\leq 1$.

Finally, on taking the principal square roots of $k^{2}$ and ${k^{\prime}}^{2}$ we obtain values for $k$ and $k^{\prime}$ that lie in the 1st and 4th quadrants, respectively, and $2\omega_{1}$, $2\omega_{3}$ are given by

where $M$ denotes the arithmetic-geometric mean (see §§19.8(i) and 22.20(ii)). This process yields 2 possible pairs ($2\omega_{1}$, $2\omega_{3}$), corresponding to the 2 possible choices of the square root.

• ##### 8: Bibliography M
• H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
• H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.