Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3 ppt

Assuming that the above expression is non-singular, we have found a formula for writing the analytic function in terms of its real part, ur, θ.. We can write a contour integral in terms

9.4 Exercises

Exercise 9.1

Consider two functions, f (x, y) and g(x, y) They are said to be functionally dependent if there is a an h(g) such that

f (x, y) = h(g(x, y))

f and g will be functionally dependent if and only if their Jacobian vanishes

If f and g are functionally dependent, then the derivatives of f are

fx = h0(g)gx

fy = h0(g)gy.Thus we have

∂(f, g)

∂(x, y) =

fx fy

gx gy

3 ex(sin x cos y cosh y − cos x sin y sinh y)

ux+ ıvx uy+ ıvy

≤

Z b a

|f (x)| |dx| ≤ (b − a) max

a≤x≤b|f (x)|

Now we prove the analogous result for the modulus of a contour integral.

Z

C

f (z) dz

=

...

f (z) dz

≤ Z

C

|f (z)| |dz| ≤

 maxz∈C |f (z)|

 (length... contours {Γn} then the integrals along {Cm} and {Γn} are equal.