The document provides an overview of Gaussian quadrature formulas, which transform integrals into linear forms using specified points. It explains the one-point, two-point, and three-point Gaussian quadrature formulas along with examples of their application. The formulas facilitate the evaluation of integrals in complex variables and numerical methods.

1. Gaussian Quadrature
Formula
-BY GROUP NO. 9
-MECHANICAL DEPARTMENT
-4TH SEMESTER
-COMPLEX VARIABLES AND NUMERICAL
METHODS(2141905)

2. Gaussian Quadrature Formula
   
 

b
a
1
1-
f(x)dx.f(t)dt to
integralthefromdtransforme
becan;
2
1
2
1
t
ofnnsformatiolinear traThe
abxab

3. Gaussian One Point Formula
1. The Gaussian One Point Formula to convert an integral into its
linear transform is given by


1
1
)0(2)( fdxxf

4. Gaussian Two Point Formula
2. The Gaussian Two Point Formula to convert an integral into its
linear transform is given by












 3
1
3
1
)(
1
1
ffdxxf

5. Gaussian Three Point Formula
3. The Gaussian Three Point Formula to convert an integral into
its linear transform is given by
 
























1
1
5
3
5
3
9
5
)0(
9
8
)( fffdxxf

6. Example of Gaussian Quadrature
Formula
formula.pointthreeand
pointtwopoint,oneby
x1
dx
Evaluate
1
1
2

2
1
1
x
Here f(x)



7. Example of Gaussian Quadrature
Formula
mulaPoint Forussian Oneound by Gawhich is f
2f(x)dx
(0)1
1
2
2f(0)f(x)dx
1
1
2
1
1















8. Example of Gaussian Quadrature
Formula

























1
1
50.1)( dxxf
1.50
0.750.75
f(0.57735)0.57735)f(
3
1
f
3
1
ff(x)dx
t Formula,n Two PoinBy Gaussia
1
1

9. Example of Gaussian Quadrature
Formula
   
 0.75
9
5
9
8
0.3750.375
9
5
1
9
8
f(x)dxNow,
0.375
5
3
0.375, f
5
3
-1, ff(0)
5
3
f
5
3
f
9
5
f(0)
9
8
f(x)dx
a,int Formuln Three PoBy Gaussia
1
-1
1
1















































10. Example of Gaussian Quadrature
Formula
1.305f(x)dx
1.305
9
11.75
9
3.75
9
8
1
-1






11. Example of Gaussian Quadrature
Formula
  Formula.hree Pointdx with T2xxEvaluate
4
2
2
 
   
 
3tx
2)(4
2
1
t24
2
1
x
42, ba
ab
2
1
tab
2
1
Here x





12. Example of Gaussian Quadrature
Formula
 
      
   dt158ttdx2xx
158tt
3t23t2xx
3txdtf(t), dxf(x)
1t4x
1t2when x
1
1
2
4
2
2
2
22
 








13. Example of Gaussian Quadrature
Formula
 
   
   31.2
9
5
15
9
8
21.7979.403
9
5
15
9
8
f(t)dt
21.797
5
3
9.403, f
5
3
15, ff(0)
5
3
f
5
3
f
9
5
0f
9
8
f(t)dt
a,int Formuln Three PoBy Gaussia
1
1
1
1

















































14. Example of Gaussian Quadrature
Formula
 
sum.the givenevaluatedFormula we
Pointsian Threese of Gauswith the uTherefore
30.667dx2xx
30.667
9
276
9
156
9
120
4
2
2






15. Thank You!