37 real Complex Variables questions from the Engineering Mathematics bank, as asked in Indian campus drives and tech interviews. Every question has a verified answer and an AI-tutor explanation on placd — free to start.
1. What is Analytic function?
Junior
A.an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
B.a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
C.the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
D.a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do
2. Which term means: "a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series"?
A.Analytic function — a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do
B.Analytic function — a mapping given by a ratio of two linear expressions with non-zero determinant, which carries circles and lines to circles and lines
C.Analytic function — an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
D.Analytic function — a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
5. Which term means: "the pair of partial differential relations between the real and imaginary parts that a complex function must satisfy to be differentiable"?
A.Cauchy-Riemann equations — the pair of partial differential relations between the real and imaginary parts that a complex function must satisfy to be differentiable
B.Cauchy-Riemann equations — expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point
C.Cauchy-Riemann equations — the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
D.Cauchy-Riemann equations — an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
A.an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
B.a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do
C.an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
D.evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
8. Which term means: "a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do"?
A.Harmonic function — a mapping by an analytic function with non-zero derivative, which preserves both the magnitude and the sense of angles between curves
B.Harmonic function — a mapping given by a ratio of two linear expressions with non-zero determinant, which carries circles and lines to circles and lines
C.Harmonic function — a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do
D.Harmonic function — an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
A.a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
B.the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
C.a mapping by an analytic function with non-zero derivative, which preserves both the magnitude and the sense of angles between curves
D.an expansion about an isolated singularity that admits negative as well as positive powers, unlike a Taylor expansion
A.Cauchy's integral theorem — an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
B.Cauchy's integral theorem — expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point
C.Cauchy's integral theorem — the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
D.Cauchy's integral theorem — an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
A.an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
B.expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point
C.a mapping by an analytic function with non-zero derivative, which preserves both the magnitude and the sense of angles between curves
D.an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
14. Which term means: "expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point"?
A.Cauchy's integral formula — the pair of partial differential relations between the real and imaginary parts that a complex function must satisfy to be differentiable
B.Cauchy's integral formula — an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
C.Cauchy's integral formula — a mapping given by a ratio of two linear expressions with non-zero determinant, which carries circles and lines to circles and lines
D.Cauchy's integral formula — expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point
A.evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
B.an expansion about an isolated singularity that admits negative as well as positive powers, unlike a Taylor expansion
C.an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
D.a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
A.Residue theorem — evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
B.Residue theorem — expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point
C.Residue theorem — an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
D.Residue theorem — a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
A.Laurent series — the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
B.Laurent series — a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do
C.Laurent series — an expansion about an isolated singularity that admits negative as well as positive powers, unlike a Taylor expansion
D.Laurent series — an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
A.a mapping by an analytic function with non-zero derivative, which preserves both the magnitude and the sense of angles between curves
B.the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
C.an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
D.an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
23. Which term means: "an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it"?
A.Simple pole — evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
B.Simple pole — an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
C.Simple pole — an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
D.Simple pole — a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
A.a twice-differentiable real function satisfying Laplace's equation, as both the real and the imaginary part of any analytic function do
B.a mapping by an analytic function with non-zero derivative, which preserves both the magnitude and the sense of angles between curves
C.expresses the value of an analytic function at an interior point as a contour integral of the function divided by the displacement from that point
D.an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
26. Which term means: "an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value"?
A.Essential singularity — a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
B.Essential singularity — the pair of partial differential relations between the real and imaginary parts that a complex function must satisfy to be differentiable
C.Essential singularity — the contour integral of a function analytic everywhere on and inside a simple closed contour is zero
D.Essential singularity — an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
A.a complex function that is differentiable at every point of some neighbourhood, equivalently one that is locally representable by a convergent power series
B.evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
C.an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
D.an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
A.Removable singularity — an isolated singularity at which the Laurent expansion has exactly one negative power term, so multiplying by the first power of the displacement removes it
B.Removable singularity — a mapping given by a ratio of two linear expressions with non-zero determinant, which carries circles and lines to circles and lines
C.Removable singularity — an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
D.Removable singularity — an isolated singularity whose Laurent expansion carries infinitely many negative power terms, near which the function takes almost every complex value
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