Solutions 1.11.4
7
Chapter 1 Pattern Recognition
1.1 Substituting (1.1) into (1.2) and then differentiating with respect to wi we obtain
N
M
n=1
j =0
wj xj tn
n
xi = 0.
n
(1)
Re-arranging terms then gives the required result.
1.2 For the regularized sum-o

Recommended References
1. Advanced Engineering Mathematics Using
MATLAB, 2nd Edition, Thomas L. Harman,
James Dabney and Norman Richert, Brooks
Cole, 2000 (Main Textbook).
2. Numerical Methods for Engineers, 7th Edition,
Steven Chapra and Raymond Canale,

Complex Numbers
Who uses them
in real life?
Complex Numbers
Who uses them
in real life?
The navigation system in the space
shuttle depends on complex numbers!
Can you see a problem here?
Who goes first?
Complex numbers do not have order
What is a complex

Transformations
Outline
Intro to Transformations
Classes of Transformations
Representing Transformations
Combining Transformations
What is a Transformation?
Maps points (x, y) in one coordinate system to
points (x', y') in another coordinate system
x' =

Solution of Differential equations
Solution of Differential equations
Power series method
Bessels equation by Frobenius method
Power series method
Method for solving linear differential equations with
variable coefficients.
y ' p ( x ) y ' q ( x ) y f

Solving Differential Equations
Using
Laplace Transform Methods
Laplace Transform
Laplace transform was developed by the French
mathematician by the same name (1749-1827).
was widely adapted to engineering problems
in the last century. Its utility lies i