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Category: Integral

Multiply 3×3 matrix

Multiply 3×3 matrix

Input
Times[Array[a, {3, 3}]]
Result
(a(1, 1) | a(1, 2) | a(1, 3)
a(2, 1) | a(2, 2) | a(2, 3)
a(3, 1) | a(3, 2) | a(3, 3))
Characteristic polynomial
lambda^2 a(3, 3) + lambda^2 a(2, 2) + lambda^2 a(1, 1) – lambda a(2, 2) a(3, 3) – lambda a(1, 1) a(3, 3) + lambda a(2, 3) a(3, 2) + lambda a(1, 3) a(3, 1) – lambda a(1, 1) a(2, 2) + lambda a(1, 2) a(2, 1) + a(1, 1) a(2, 2) a(3, 3) – a(1, 2) a(2, 1) a(3, 3) – a(1, 1) a(2, 3) a(3, 2) + a(1, 3) a(2, 1) a(3, 2) + a(1, 2) a(2, 3) a(3, 1) – a(1, 3) a(2, 2) a(3, 1) – lambda^3
Mathematica limit

Mathematica limit

Input interpretation
Limit (Wolfram Language symbol)
Usage
Limit[expr, x->x0] finds the limiting value of expr when x approaches x0.
Basic example
In[1]:=Limit[Sin[x]/x, x->0]
Out[1]=1
In[2]:=Limit[(1 + x/n)^n, n->Infinity]
Out[2]=e^x
Options
Assumptions | Direction
Common option values
Direction | 1 | -1
Attributes
Listable | Protected | ReadProtected
Relationships with other entities
Series | Residue | Derivative | Assumptions | DiracDelta | PrincipalValue
Integral of sinx/x

Integral of sinx/x

Indefinite integral
integral (sin(x))/x dx = Si(x) + constant
Series expansion of the integral at x=0
x – x^3/18 + x^5/600 + O(x^6)
(Taylor series)
Series expansion of the integral at x=∞
sin(x) (-(1/x)^2 + 6/x^4 + O((1/x)^6)) + cos(x) (-1/x + 2/x^3 – 24/x^5 + O((1/x)^6)) + pi/2
Definite integral
integral_0^pi (sin(x))/x dx = Si(pi)~~1.85194
Y=x^2+1

Y=x^2+1

Input
y = x^2 + 1
Geometric figure
parabola
Alternate form
-x^2 + y – 1 = 0
Derivative
(d)/(dx)(x^2 + 1) = 2 x
Global minimum
min{x^2 + 1} = 1 at x = 0
9x-7i 3(3x-7u)

9x-7i 3(3x-7u)

Input
9 x – (7 i) (3 (3 x – 7 u))
Result
9 x – 21 i (3 x – 7 u)
Alternate form assuming u and x are real
9 x + i (147 u – 63 x)
Root
x = (343/150 – (49 i)/150) u
Property as a function
odd
Root for the variable u
u = (3/7 + (3 i)/49) x
Derivative
(d)/(dx)(9 x – (7 i) (3 (3 x – 7 u))) = 9 – 63 i
Indefinite integral
integral (9 x – (7 i) (3 (3 x – 7 u))) dx = 147 i u x + (9/2 – (63 i)/2) x^2 + constant
Definite integral over a disk of radius R
integral integral_(u^2 + x^2
Definite integral over a square of edge length 2 L
integral_(-L)^L integral_(-L)^L (9 x – 21 i (-7 u + 3 x)) dx du = 0