Complex Variables
Contents
Problems
solution Evaluate
solution If Failed to parse (unknown error):
Find
solution
solution
solution Show that the four points in the Argand plane represented by the complex numbers are the vertices of a square Failed to parse (unknown error):
solution Find the equation of the straight line joining the points
in the Argand plane
solution Evaluate .
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution If Failed to parse (unknown error):
, then what is Failed to parse (unknown error):
?
solution If Failed to parse (unknown error):
, then what is Failed to parse (unknown error):
?
solution If Failed to parse (unknown error):
, then what is Failed to parse (unknown error):
?
solution If Failed to parse (unknown error):
find Failed to parse (unknown error):
.
solution If Failed to parse (unknown error):
find Failed to parse (unknown error):
.
solution If Failed to parse (unknown error):
find Failed to parse (unknown error):
.
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution List all the cube roots of Failed to parse (unknown error):
solution List all the cube roots of Failed to parse (unknown error):
solution If Failed to parse (unknown error):
find Failed to parse (unknown error): and check if it satisfies the Cauchy-Riemann equations.
Differentiation
solution Show that Failed to parse (unknown error):
is non-analytic everywhere.
solution Find Failed to parse (unknown error):
.
solution Show that Failed to parse (unknown error):
solution Show that Failed to parse (unknown error):
solution Show that if Failed to parse (unknown error):
is harmonic then Failed to parse (unknown error): is analytic.
Polynomials
solution Find the partial fraction decomposition of Failed to parse (unknown error):
.
solution Deflate Failed to parse (unknown error):
solution Show that a polynomial with real coefficients can always be expressed as a product of linear and quadratic factors with real coefficients.
solution Write the Taylor expansion of Failed to parse (unknown error):
at Failed to parse (unknown error):
.
solution Write the Taylor expansion of Failed to parse (unknown error):
at Failed to parse (unknown error):
.
solution Write the Taylor expansion of Failed to parse (unknown error):
at Failed to parse (unknown error):
.
Trigonometric Functions
solution Verify the identity: Failed to parse (unknown error):
solution Verify the identity: Failed to parse (unknown error):
solution Verify the identity: Failed to parse (unknown error):
solution Verify the identity: Failed to parse (unknown error):
solution Verify the identity: Failed to parse (unknown error):
Exponential and Log
This site uses Failed to parse (unknown error):
and Failed to parse (unknown error): for the principal values.
solution Evaluate Failed to parse (unknown error):
solution Find the domain of analyticity for Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution Find where Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
on the principal branch.
solution Evaluate Failed to parse (unknown error):
on the principal branch.
solution Evaluate Failed to parse (unknown error):
on the principal branch.
solution Solve Failed to parse (unknown error):
Find harmonic functions on certain domains
solution Find a function that is harmonic on the washer-shaped region between the circles Failed to parse (unknown error):
and takes the values 20 and 30 on the inner and outer circles.
solution Find a function that is harmonic on the wedge-shaped region between the rays in the complex plane with principal argument Failed to parse (unknown error):
and takes the values 20 and 30 on rays with the smaller and larger angles.
solution Find a function that is harmonic on the vertical strip from Failed to parse (unknown error): 1 to 2 and equals 20 and 30 at Failed to parse (unknown error): 1 and 2.
solution Find a function that is harmonic on the washer-shaped region between the circles with radii 1 and 2 and center Failed to parse (unknown error): . It should be 0 and 10 on the inner and outer circle.
solution Find a function that is harmonic on the strip between the lines Failed to parse (unknown error):
that takes the values -50 and 10 on the lower an upper lines.
Series
solution Find the Laurent series for about the singular point Failed to parse (unknown error): .
solution Find the Maclaurin series for Failed to parse (unknown error): .
solution Find the Laurent series for Failed to parse (unknown error):
about all its singular points.
solution Find the Laurent series about Failed to parse (unknown error):
for the function Failed to parse (unknown error):
Residues
Find the residues of Failed to parse (unknown error):
at all its isolated singular points and at infinity (if infinity is not a limit point of singular points), where Failed to parse (unknown error): is given by
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
Complex Integrals
solution Give an upper bound for Failed to parse (unknown error):
solution Compute Failed to parse (unknown error):
along the directed line segment from Failed to parse (unknown error): to
Failed to parse (unknown error): .
solution Evaluate Failed to parse (unknown error):
where Failed to parse (unknown error): is the circle traversed twice in the clockwise direction.
solution Evaluate .
solution Evaluate .
solution Give an upper bound for is the line segment from Failed to parse (unknown error):
to 1.
solution Evaluate Failed to parse (unknown error):
starts at the origin, traverses the bottom half of a unit circle centered at Failed to parse (unknown error): and then the line from to .
Contour Integrals
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
solution Failed to parse (unknown error):
Residue Calculus
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution Evaluate Failed to parse (unknown error):
solution Prove that Failed to parse (unknown error):
.
solution A function Failed to parse (unknown error):
is zero when Failed to parse (unknown error):
, and is real when Failed to parse (unknown error):
is real, and is analytic when Failed to parse (unknown error): . If Failed to parse (unknown error):
is the imaginary part of Failed to parse (unknown error): prove that Failed to parse (unknown error): holds when Failed to parse (unknown error):
.
solution Prove that Failed to parse (unknown error): . Hint: Integrate Failed to parse (unknown error):
over the semicircle contour in the upper half plane, then put Failed to parse (unknown error):
.
solution Show that Failed to parse (unknown error): , if Failed to parse (unknown error):
are real, Failed to parse (unknown error): is positive and Failed to parse (unknown error):
.
solution Evaluate Failed to parse (unknown error):
Proofs
solution Show that Failed to parse (unknown error):
.
solution Show that Failed to parse (unknown error):
.
solution Show that Failed to parse (unknown error):
.
solution Show that Failed to parse (unknown error):
.
solution Show that Failed to parse (unknown error):
Facts
- The Failed to parse (unknown error):
roots of a complex number written in polar form Failed to parse (unknown error): are
Failed to parse (unknown error):
De Moivre's Theorem
If Failed to parse (unknown error): and Failed to parse (unknown error): then:
Failed to parse (unknown error):
Failed to parse (unknown error):
Prove it by induction.
- Failed to parse (unknown error):
- For every complex number Failed to parse (unknown error):
and any positive integer Failed to parse (unknown error):
, it is true that
Failed to parse (unknown error):
- Every subset of the complex plane is compact if and only if it is closed and bounded.
- The complement of an open set is closed and vice versa.
- If Failed to parse (unknown error):
is continuous at Failed to parse (unknown error):
, then it must be true that Failed to parse (unknown error):
.
- The function Failed to parse (unknown error):
is one-to-one and continuous everywhere on the complex plane except at Failed to parse (unknown error):
.