Complex Analysis
| Description: This quiz covers various concepts and techniques in Complex Analysis. | |
| Number of Questions: 15 | |
| Created by: Aliensbrain Bot | |
| Tags: complex analysis complex numbers analytic functions cauchy's theorem residue theorem |
Which of the following is the imaginary unit?
-
$i$
-
$\pi$
-
$\sqrt{-1}$
-
$\infty$
The imaginary unit, denoted by $i$, is defined as the square root of -1, i.e., $i = \sqrt{-1}$. It is used to represent imaginary numbers, which are numbers that have a real part of 0.
What is the Cauchy-Riemann equation in complex analysis?
-
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
-
$\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
-
$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
-
$\frac{\partial u}{\partial x} = -\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$
The Cauchy-Riemann equation is a system of two partial differential equations that are necessary and sufficient conditions for a complex function to be differentiable at a point.
What is the residue of a function at a pole?
-
The value of the function at the pole
-
The coefficient of the highest power of $z - a$ in the Laurent expansion of the function around the pole $a$
-
The integral of the function around a small circle centered at the pole
-
The derivative of the function at the pole
The residue of a function at a pole is a complex number that measures the behavior of the function near the pole.
Which of the following is a consequence of Cauchy's Integral Theorem?
-
The fundamental theorem of calculus
-
The residue theorem
-
The maximum modulus principle
-
Liouville's theorem
Cauchy's Integral Theorem is a fundamental result in complex analysis that relates the value of a complex function at a point to the integral of the function around a closed contour.
What is the maximum modulus principle?
-
A theorem that states that the maximum value of a continuous function on a closed and bounded set is attained at a boundary point.
-
A theorem that states that the maximum value of an analytic function on a closed and bounded set is attained at a boundary point.
-
A theorem that states that the maximum value of a harmonic function on a closed and bounded set is attained at a boundary point.
-
A theorem that states that the maximum value of a subharmonic function on a closed and bounded set is attained at a boundary point.
The maximum modulus principle is a fundamental result in complex analysis that provides a bound on the maximum value of an analytic function on a closed and bounded set.
What is the argument principle?
-
A theorem that relates the number of zeros of an analytic function inside a closed contour to the change in the argument of the function around the contour.
-
A theorem that relates the number of poles of an analytic function inside a closed contour to the change in the argument of the function around the contour.
-
A theorem that relates the number of zeros and poles of an analytic function inside a closed contour to the change in the argument of the function around the contour.
-
A theorem that relates the number of zeros of an analytic function outside a closed contour to the change in the argument of the function around the contour.
The argument principle is a fundamental result in complex analysis that relates the number of zeros of an analytic function inside a closed contour to the change in the argument of the function around the contour.
What is the residue theorem?
-
A theorem that relates the value of a complex function at a point to the integral of the function around a closed contour.
-
A theorem that relates the value of a complex function at a pole to the integral of the function around a small circle centered at the pole.
-
A theorem that relates the value of a complex function at a zero to the integral of the function around a small circle centered at the zero.
-
A theorem that relates the value of a complex function at a singularity to the integral of the function around a small circle centered at the singularity.
The residue theorem is a fundamental result in complex analysis that relates the value of a complex function at a pole to the integral of the function around a small circle centered at the pole.
What is the order of a pole of a complex function?
-
The degree of the denominator of the function at the pole
-
The degree of the numerator of the function at the pole
-
The degree of the function at the pole
-
The degree of the derivative of the function at the pole
The order of a pole of a complex function is the degree of the denominator of the function at the pole.
What is the Laurent expansion of a complex function around a point?
-
An infinite series representation of the function in terms of powers of $z - a$
-
An infinite series representation of the function in terms of powers of $z$
-
An infinite series representation of the function in terms of powers of $1/z$
-
An infinite series representation of the function in terms of powers of $1/(z - a)$
The Laurent expansion of a complex function around a point is an infinite series representation of the function in terms of powers of $z - a$.
What is the principle of analytic continuation?
-
A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still analytic.
-
A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still continuous.
-
A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still differentiable.
-
A principle that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still integrable.
The principle of analytic continuation is a fundamental principle in complex analysis that states that an analytic function can be extended to a larger domain by finding a new domain where the function is still analytic.
What is the Riemann mapping theorem?
-
A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the unit disk.
-
A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the upper half-plane.
-
A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the lower half-plane.
-
A theorem that states that every simply connected open set in the complex plane can be conformally mapped onto the right half-plane.
The Riemann mapping theorem is a fundamental result in complex analysis that states that every simply connected open set in the complex plane can be conformally mapped onto the unit disk.
What is the Schwarz-Christoffel formula?
-
A formula that gives the conformal mapping of a polygon onto the upper half-plane.
-
A formula that gives the conformal mapping of a polygon onto the unit disk.
-
A formula that gives the conformal mapping of a polygon onto the lower half-plane.
-
A formula that gives the conformal mapping of a polygon onto the right half-plane.
The Schwarz-Christoffel formula is a formula that gives the conformal mapping of a polygon onto the upper half-plane.
What is the Weierstrass factorization theorem?
-
A theorem that states that every entire function can be written as a product of a sequence of elementary functions.
-
A theorem that states that every meromorphic function can be written as a product of a sequence of elementary functions.
-
A theorem that states that every analytic function can be written as a product of a sequence of elementary functions.
-
A theorem that states that every harmonic function can be written as a product of a sequence of elementary functions.
The Weierstrass factorization theorem is a fundamental result in complex analysis that states that every entire function can be written as a product of a sequence of elementary functions.
What is the Mittag-Leffler expansion theorem?
-
A theorem that states that every meromorphic function can be written as a sum of a series of partial fractions.
-
A theorem that states that every analytic function can be written as a sum of a series of partial fractions.
-
A theorem that states that every harmonic function can be written as a sum of a series of partial fractions.
-
A theorem that states that every entire function can be written as a sum of a series of partial fractions.
The Mittag-Leffler expansion theorem is a fundamental result in complex analysis that states that every meromorphic function can be written as a sum of a series of partial fractions.
What is the Runge approximation theorem?
-
A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of rational functions.
-
A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of polynomials.
-
A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of trigonometric functions.
-
A theorem that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of exponential functions.
The Runge approximation theorem is a fundamental result in complex analysis that states that every continuous function on a closed and bounded set in the complex plane can be uniformly approximated by a sequence of rational functions.