Thanks. ]]>

fast to comprehend. I congratulate you because of this research that I

am going to recommend to the people friends. I request you to visit the gpa-calculator.co page where each university student or university student can find results gpa marks.

Success! ]]>

LAGRANGIAN AND PARTIAL DIFFERENTIAL DERIVATIVES ]]>

The students ought to go six hours every week

underneath the supervision of an veterinary and he have to keep the journal about

the numerous expertise that he’s acquired before attending the institution https://math-problem-solver.com/ .

Felleisen’s role on this course of.

f(z) = u(x,y) + iv(x,y)

in which v and u are real. For example if f(z)=z2, we have

The relationship between z and f(z) is best pictured as a mapping operation, we address it in detail later.

Complex Conjugation: replacing I by –I, which is denoted by (*),

We then have

]]>The slope of the function at x0 is then

(x0) = (x0) + (x0) = [ f(x1) f(x0)] + (x0)

Forward difference approximation is obtained when the slope of the interpolating polynomial estimates the derivative of the function at x0 as shown in Figure 3.3-2. In term of the forward difference operator

(x0) = + (x0)

where f(xi) = f(xi+1) f(xi).

Figure 3.3-2 Derivatives of the function at x0 and at x1.

The error for the derivative can be estimated by taking derivative of the error

E1(x) = ()

(x0) = [ (x x0)(x x1) ()

(x0) = () [(x x0)(x x1)

(x0) = () [(x x1) + (x x0) = h () = O(h)

Backward difference approximation is obtained when the slope of the interpolating polynomial estimates the derivative of the function at x1 as shown in Figure 3.3-2.

(x1) = (x1) + (x1) = [ f(x1) f(x0)] + (x1)

The error term has the form

(x1) = () [(x x1) + (x x0) = h () = O(h)

The error in the backward difference approximation, while having the same form as that in the forward difference approximation, has a different sign. In term of the backward difference operator

Figure 3.3-3 Derivative of the function at x1 is estimated by a second-degree polynomial.

Let h = x1 x0 = x2 x1, the three points interpolating polynomial over this interval is

P2(x) = L2,0(x) f(x0) + L2,1(x)f(x1) + L2,2(x)f(x2)

P2(x) = f(x0) + f(x1) + f(x2)

P2(x) = [(x x1)(x x2)f(x0) 2(x x0)(x x2)f(x1) + (x x0)(x x1) f(x2)]

The function f(x) can be expressed in terms of its approximating polynomial with an error as

f(x) = P2(x) + E2(x)

The slope of the function at x1 is then

(xi) = + O(h)

where f(xi) = f(xi) f(xi-1)

Central difference approximation is obtained when the slope of the interpolating polynomial estimates the derivative of the function at the midpoint x1 as shown in Figure 3.3-3.

P1(x) = f(x0) + f(x1)

Let h = (x1 x0), then

P1(x) = [(x1 x) f(x0) + (x x0) f(x1)]

Figure 3.3-1 Approximating by first-degree polynomial with error E1(x).

The function f(x) can be expressed in terms of its approximating polynomial with an error as

f(x) = P1(x) + E1(x) (x1) = (x1) + (x1) = [ f(x2) f(x0)] + (x1)

The error term in this difference approximation is

(x1) = [ (x x0)(x x1)(x x2) ()

(x1) = h2 () = O(h2)

The three points approximation is accurate to O(h2). In term of the central difference operator

(x1) = [f(x1 + ) + f(x1 )] + O(h2)

where the central difference operator is defined as

f(xi) = f(xi + ) (xi )

f(x1 + ) + f(x1 ) = f(x1 + h) f(x1) + f(x1) f(x1 h) = f(x2) f(x0)

Finite difference approximation of higher order derivatives can also be obtained(x) = P2(x) + E2(x)

The second derivative of the function at x1 is then

(x1) = (x1) + (x1)

After some algebra

(x1) = [f(x0) 2f(x1) + f(x2)] + (x1)

The error term can be evaluated to yield

(x1) = h2 () = O(h2)

The central difference can also be written in terms of the central difference operator

(x1) = + O(h2)

where

2f(x1) = [f(x1)] = [ f(x1 + ) f(x1 )] = f(x2) f(x1) [f(x1) f(x0)]

So we have INTEGRATION , Prof. Orasanu

And appear some

]]>