LINE INTEGRAL & ML INEQUALITY

140410119100 - Jay Rami140410119101 - Malhar Rana140410119102 –Raval Harsh140410119103 – Raval Parth140410119104 – Ray Bhargav

LINE INTEGRAL Let f(z)be a continuous function of the

complex variable z=x+iy defined at every point of a curve c whose end points are A and B divide the curve c into n parts at the points

A=P0(z0),P1(z1)……..Pn(zn)=B Line integral of f(z) along the path C and

is denoted by dz if c is a closed curve.

LINE INTEGRALSuppose that the equation z=z(t)

represents the contour c from point z=a to z=b also f(z) is piecewise continuous on c then line integral or contour integral of f

along c in terms of parameter of t is : c .z’(t)dt

Provided z’(t) is piecewise continuous.

PROPERTIES OF LINE INTEGRALS:- If F(z) and G(z) are integrable along a curve C then the following properties hold:

1. Linearity2. Sense reversal 3. Partitioning of path 4. Integral inequality5. ML inequality

1. Linearity:-c+k2 G(z)]dz=k1 cdz +k2 cdzFor example:

2. Sense reversal:- dz=- dz3.Patitioning of path:-

if the curve C consists of the curve c1 and c2 then:dz+c2 dz4. Integral inequality:

|c dz|=<

5. M L inequality:-if F(Z)is continuos on the curve c of

length L and |f(Z)|=<M then:|c dz|=<ML

Thank

you