37 Complex Variables questions from the Engineering Mathematics bank, written for Indian campus drives and tech interviews. Every question has a verified answer and an AI-tutor explanation on placd.
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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
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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"?
Junior
A.Bilinear transformation
B.Residue theorem
C.Analytic function
D.Conformal mapping
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3. Which statement is correct?
Junior
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
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4. What is Cauchy-Riemann equations?
Junior
A.an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there
B.the pair of partial differential relations between the real and imaginary parts that a complex function must satisfy to be differentiable
C.evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
D.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
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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"?
Junior
A.Analytic function
B.Removable singularity
C.Cauchy-Riemann equations
D.Residue theorem
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6. Which statement is correct?
Junior
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
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7. What is Harmonic function?
Junior
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
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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"?
Junior
A.Harmonic function
B.Bilinear transformation
C.Laurent series
D.Cauchy's integral theorem
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9. Which statement is correct?
Junior
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
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10. What is Cauchy's integral theorem?
Mid
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
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11. Which term means: "the contour integral of a function analytic everywhere on and inside a simple closed contour is zero"?
Mid
A.Cauchy's integral theorem
B.Laurent series
C.Analytic function
D.Simple pole
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12. Which statement is correct?
Mid
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
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13. What is Cauchy's integral formula?
Mid
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
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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"?
Mid
A.Residue theorem
B.Cauchy's integral formula
C.Cauchy's integral theorem
D.Laurent series
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15. Which statement is correct?
Mid
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
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16. What is Residue theorem?
Mid
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
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17. Which term means: "evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour"?
Mid
A.Cauchy-Riemann equations
B.Residue theorem
C.Harmonic function
D.Essential singularity
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18. Which statement is correct?
Mid
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
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19. What is Laurent series?
Mid
A.the pair of partial differential relations between the real and imaginary parts that a complex function must satisfy to be differentiable
B.evaluates a closed contour integral as two-pi-i times the sum of the residues at the singularities enclosed by the contour
C.a mapping given by a ratio of two linear expressions with non-zero determinant, which carries circles and lines to circles and lines
D.an expansion about an isolated singularity that admits negative as well as positive powers, unlike a Taylor expansion
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20. Which term means: "an expansion about an isolated singularity that admits negative as well as positive powers, unlike a Taylor expansion"?
Mid
A.Bilinear transformation
B.Essential singularity
C.Laurent series
D.Cauchy's integral formula
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21. Which statement is correct?
Mid
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
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22. What is Simple pole?
Mid
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
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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"?
Mid
A.Cauchy's integral formula
B.Conformal mapping
C.Simple pole
D.Residue theorem
Answer + AI explanation with Pro
24. Which statement is correct?
Mid
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
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25. What is Essential singularity?
Senior
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
Answer + AI explanation with Pro
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"?
Senior
A.Cauchy's integral theorem
B.Removable singularity
C.Simple pole
D.Essential singularity
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27. Which statement is correct?
Senior
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
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28. What is Removable singularity?
Senior
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
Answer + AI explanation with Pro
29. Which term means: "an isolated singularity at which the function stays bounded, so redefining a single value makes the function analytic there"?
Senior
A.Removable singularity
B.Conformal mapping
C.Analytic function
D.Laurent series
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30. Which statement is correct?
Senior
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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