AS2050
Tutorial
01 (Eriday
DrobleuS Withh
01 August
2025)
students.
vour follow
so ou
Iike typo and
Correct Distakes
to
fuc
problems. if needed; and state
calefullr aid fecl
the problens
help solve the
a s s u m p t i o n s to
V Pcase rcad
the.
to make nccessary
v Please fol frce
introduce and justify
problems.
assnptions you
solve the
the
CYplctiy
should be uscd to
Course
the
possiblC.
taught in
sketches when
V The materials
physical rea
problems, please use
the
solve
and operations and
mathematical equations
To help visualize and
present
details' means that you
Techinical
V o u ate
cncoutagcd
to discuSs the
sons/ justihcations underlying.
1. Review the materials
2. Solve
covercd and list those that
are new to you.
the solutions in the
z = 1. Skctch adcquately
3. Complex Stokes' theorem
z-plane and describe the
z), continuous and
says that for a function f(z.
geomctric pattern.
differentiable in the
area
S encloscd bv C.
= -2i
Js
Derive the above equation with the help of
the Stokes' theoren for real functions
| a. y) drtilr. y) dy -
drdy.
Oy
JS
Hint: (i) U, =Ref(:. ¿), , = 0: (ii) v, =0, Vy = Imf(z, ~): (ii)
v, and v:
=
+
4. We state Cauchv's residue theorem below.
Let C be aclosed contour inside and upon which the function f(z) is holonorphic, except
at a finite number of singular points
are aj. ay.
2,: i= 1,:: ,n} within Cat which the residues
, ay. Then
|fe) dz = 2ri(a, +ag t
+an).
(a) Regarding the f(z) described above, give one example of your own.
(b) We have discussed the simplest version of the theorem in lecture involving single pole (n = 1).
How do you extend the ideas like cuts presented in lecture to prove the general version? Please
give technical details. (Hint: Consider n =2 first?\
(c) Try to give one example of yours toapply the theorem?
5. Verify that, under the transformation of ( = + in = f(z) where f is
analytic,
0 in the 2-plane implies
Here,
is continuous and as SImooth as desired/required.
6. Imagjine that you were an instructor in
+
=0in the (-plane.
Hint: and n are conjugate; chain rule.)
this
the students' grasping and understanding, course. Come up with one qucstion of your own to test
based on tlhe materials covercd so far.