Math 303 - Complex Variables
Homework due March 4
Note: In the following, “FTC” means the Fundamental Theorem of Calculus.
Question 1. Compute the following complex integral in two ways: (i) writing f(t) = u(t) + iv(t) and
taking two separate integrals by using the FTC twice and (ii) taking the antiderivative of f(t) (without
splitting it into the real and imaginary parts) and using the FTC once. Do you get the same answer?
Z1
0
(3t−i)2dt
Question 2.
(a) Find Zπ /2
0
et+it dt by writing
et+it =et·eit =et(cos t+isin t) = etcos t+ietsin t
and computing the integral of the real and imaginary part separately.
(b) If f(t) = et+it , find an antiderivative F(t). [Hint: et+it =et(1+i)]
(c) Use (b) and a single use of the FTC to compute the complex integral Zπ/2
0
et+it dt. Did you get
the same answer as in (a)?
Notice: In (a), you had to perform two integration by parts which were individually messy (each took
a double IBP). Clever Mathematicians and other Scientists prefer to use the methodology in (c) (with
the single use of FTC) because the real part of the answer you got in (c) equals the integral
Zπ/2
0
etcos t dt
without having to do integration by parts! So, when faced with a difficult real-valued integral
Rb
au(t)dt, clever scientists frequently find a complex-valued function f(t) such that u(t) is its real part
and proceed by finding an easier antiderivative.
Question 3. Let mand nbe integers. Use a single FTC to show that
Z2π
0
eimte−int dt =0 when m6=n
2πwhen m=n.
By showing this, you have shown that eint forms a family of orthonormal eigenfunctions that are used
in Fourier Analysis.
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