Math 656 March 10, 2011
Midterm Examination
This is a closed-book exam; neither notes nor calculators are allowed. Explain your work
Note: points add up to 108. You only need 100 points.
1) (14pts) Derive the expression for sinh1 z (arcsinh z) using the d
Math 656 March 10, 2011
Midterm Examination Solutions
1) (14pts) Derive the expression for sinh1 z (arcsinh z) using the definition of sinh w in terms of
exponentials, and use it to find all values of sinh1(2i). Plot these values as points in the complex
Math 656 FINAL EXAM May 11, 2010
This is a closed-book exam; neither notes nor electronic devices are allowed. Please explain all work.
1) (20pts) Categorize all zeros and singularities of the following functions, find two lowest-order non-zero
terms in t
Math 665 FINAL EXAM May 13, 2010
1) Categorize all zeros and singularities of the following functions, find two lowest-order non-zero terms in the
Laurent or Taylor series of f(z) near the given point zo, and state the region on which the corresponding
ex
Math 656 * Homework 20
Due Thursday April 21, 2011
dx
1 x
1. Calculate
6
Integrand has 3 simple poles in upper half-plane:
3 i
;i
z1,2,3 (1)1/6 (ei i 2 k )1/6 cfw_ei /6 , i, ei 5 /6
2
Method 1 Closing via semi-circle CR in the upper half-plane:
dx
dz
Math 656 * Homework 19
Due Monday April 18, 2011
1. Calculate the following improper integrals:
+
(c)
x
0
dx
(assume a > 0, otherwise intergal does not converge)
5
+ a5
Close the contour along the boundary of circular sector with angle 2 / 5 :
C = Cx + C
Math 656 * Homework 17
Due Thursday April 7, 2011
1. The proof that a function is uniquely defined by its values on any set in D which has a
limit point in D, follows directly from the proof of the Identity Theorem, which says
that only a function which i