Which condition must hold for limz→af(z)limz→af(z) to exist in the complex plane
A Same on real axis
B Same on imaginary axis
C Same along all paths
D Only bounded near aa
In complex limits, zz can approach aa from infinitely many directions. The limit exists only if f(z)f(z) approaches one unique value independent of the path taken.
If two different approach paths give different values, what can be concluded about the limit
A Limit does not exist
B Limit equals average
C Limit is infinite
D Limit exists at infinity
A complex limit must be path independent. If approaching aa along two paths yields different values, then no single number satisfies the definition of the limit.
What does ∣z∣→∞∣z∣→∞ mean for a complex variable
A Real part goes to zero
B Magnitude grows without bound
C Imaginary part stays fixed
D Argument becomes zero
The statement ∣z∣→∞∣z∣→∞ means the distance of zz from the origin becomes arbitrarily large, regardless of direction in the complex plane.
The substitution often used for limits at infinity is
A w=z+1w=z+1
B w=zˉw=zˉ
C w=z2w=z2
D w=1/zw=1/z
To study behavior as ∣z∣→∞∣z∣→∞, set w=1/zw=1/z. Then ∣z∣→∞∣z∣→∞ becomes w→0w→0, making the limit easier to evaluate.
If f(z)=z3f(z)=z3 then as ∣z∣→∞∣z∣→∞, f(z)f(z) behaves like
A Grows without bound
B Tends to zero
C Stays bounded
D Becomes periodic
For a polynomial znzn with n>0n>0, the magnitude ∣zn∣=∣z∣n∣zn∣=∣z∣n increases without limit as ∣z∣→∞∣z∣→∞.
If a rational function has higher degree in denominator, its limit at infinity is usually
A One
B Infinity
C Zero
D Undefined always
For f(z)=P(z)/Q(z)f(z)=P(z)/Q(z), if degQ>degPdegQ>degP, then ∣f(z)∣∣f(z)∣ decreases like 1/∣z∣k1/∣z∣k as ∣z∣→∞∣z∣→∞, so the limit becomes 00.
Continuity at a point aa in complex domain means
A limz→af(z)=f(a)limz→af(z)=f(a)
B f(a)=0f(a)=0
C f(z)f(z) is constant
D f(z)f(z) has no zeros
The definition of continuity in CC matches real analysis: the function is continuous at aa if the limit exists and equals the function value there.
Which class of functions is always continuous on CC
A zˉzˉ only
B 1/(z−a)1/(z−a) at aa
C argzargz everywhere
D Polynomials
Polynomial functions are built from addition and multiplication, which preserve continuity. Hence polynomials are continuous for all complex zz.
A rational function is continuous at points where
A Numerator is nonzero
B Argument is constant
C Denominator is nonzero
D It is multivalued
P(z)/Q(z)P(z)/Q(z) is continuous wherever Q(z)≠0Q(z)=0. At zeros of QQ, the function is not defined and continuity fails there.
A complex function is commonly written as
A u(x,y)+iv(x,y)u(x,y)+iv(x,y)
B u(x)+v(y)u(x)+v(y)
C u(z)+v(z)u(z)+v(z)
D u+vu+v only
With z=x+iyz=x+iy, a complex function f(z)f(z) is expressed via real and imaginary parts: f(z)=u(x,y)+i v(x,y)f(z)=u(x,y)+iv(x,y), where u,vu,v are real-valued functions.
What is the domain of a complex function
A Set of output values
B Set of allowed zz
C Only real numbers
D Only magnitudes
The domain is the collection of complex inputs zz for which the function is defined. The range is the set of complex values produced by those inputs.
Which function is typically multivalued in complex analysis
A z2z2
B ezez
C logzlogz
D z+1z+1
The complex logarithm includes infinitely many values because argzargz can differ by 2πk2πk. So logz=ln∣z∣+i(\Argz+2πk)logz=ln∣z∣+i(\Argz+2πk).
A branch cut is mainly used to make a function
A Single-valued
B Constant
C Polynomial
D Entire always
Multivalued functions like logzlogz or z1/2z1/2 become single-valued after choosing a branch and introducing a branch cut to restrict the argument range.
What does the mapping w=z+aw=z+a represent geometrically
A Rotation
B Inversion
C Reflection only
D Translation
Adding a complex constant shifts every point by the same vector. Thus w=z+aw=z+a moves the entire plane without changing shapes or angles.
The mapping w=eiθzw=eiθz represents
A Translation by θθ
B Rotation by θθ
C Inversion in circle
D Scaling by θθ
Multiplying by eiθeiθ keeps magnitude the same and increases the argument by θθ. So it rotates the point around the origin by angle θθ.
The mapping w=czw=cz with real c>0c>0 gives
A Reflection
B Shear
C Scaling
D Folding
If c>0c>0 is real, then ∣w∣=c∣z∣∣w∣=c∣z∣ and the argument stays unchanged. This stretches or shrinks distances from the origin by factor cc.
The mapping w=1/zw=1/z sends large ∣z∣∣z∣ values to
A Small ∣w∣∣w∣ values
B Larger ∣w∣∣w∣ values
C Same magnitude
D Fixed argument only
Since ∣w∣=∣1/z∣=1/∣z∣∣w∣=∣1/z∣=1/∣z∣, points far from the origin map near the origin, and points near the origin map far away (excluding z=0z=0).
The complex derivative at aa is defined using
A Area under curve
B Only real direction
C Difference quotient limit
D Only imaginary direction
f′(a)=limz→af(z)−f(a)z−af′(a)=limz→az−af(z)−f(a), if this limit exists. In complex analysis, it must be the same no matter how zz approaches aa.
A key difference from real derivatives is that complex differentiability requires
A Only continuity
B Only boundedness
C Only integrability
D Direction independence
In RR, approach is from two sides. In CC, approach is from infinitely many directions, so the derivative must be identical for all paths toward aa.
Derivative of f(z)=znf(z)=zn for integer n≥1n≥1 is
A nzn−1nzn−1
B zn+1zn+1
C nznz
D zn−1/nzn−1/n
Power rule holds for complex polynomials as well. Using the difference quotient and algebra, ddz(zn)=nzn−1dzd(zn)=nzn−1 for integers n≥1n≥1.
Derivative of ezez with respect to zz is
A zezzez
B ezez
C lnzlnz
D 1/z1/z
The complex exponential has the same series definition as real: ez=∑zn/n!ez=∑zn/n!. Differentiating termwise gives the same series back, so the derivative is ezez.
Cauchy–Riemann equations relate partial derivatives of
A xx and yy
B ∣z∣∣z∣ and argzargz
C uu and vv
D real axis only
For f(z)=u(x,y)+iv(x,y)f(z)=u(x,y)+iv(x,y), the Cauchy–Riemann equations connect uu and vv through partial derivatives, forming a key test for complex differentiability.
In Cartesian form, Cauchy–Riemann equations are
A ux=vy, uy=−vxux=vy,uy=−vx
B ux=uy, vx=vyux=uy,vx=vy
C ux=−vy, uy=vxux=−vy,uy=vx
D u=v, ux=vxu=v,ux=vx
If ff is complex differentiable and u,vu,v have suitable partial derivatives, then ux=vyux=vy and uy=−vxuy=−vx. These capture the direction-independence requirement.
If Cauchy–Riemann equations fail at a point, the function is
A Entire everywhere
B Not analytic there
C Constant near point
D Always multivalued
For standard differentiability conditions, failing the CR equations at a point means the function cannot be complex differentiable there, so it is not analytic at that point.
For f(z)=z2f(z)=z2, the function is
A Analytic only at 0
B Nowhere analytic
C Multivalued
D Analytic everywhere
z2z2 is a polynomial, and all polynomials are complex differentiable for every zz. Hence f(z)=z2f(z)=z2 is analytic on the entire complex plane.
For f(z)=zˉf(z)=zˉ, the function is
A Analytic everywhere
B Analytic only on unit circle
C Not analytic anywhere
D Analytic at infinity only
Writing zˉ=x−iyzˉ=x−iy gives u=xu=x, v=−yv=−y. Then ux=1ux=1 but vy=−1vy=−1, so CR equations fail everywhere, hence zˉzˉ is nowhere analytic.
An analytic function is also called
A Holomorphic
B Discontinuous
C Multivalued
D Merely bounded
“Analytic” in complex analysis is often used interchangeably with “holomorphic,” meaning complex differentiable in an open region. This is a stronger property than real differentiability.
An entire function means analytic on
A Only a line
B All complex plane
C Only unit disk
D Only outside disk
Entire functions are analytic for every complex number. Classic examples include polynomials, ezez, sinzsinz, and coszcosz, each analytic on CC.
A rational function is analytic except at its
A Zeros only
B Constant points
C Poles
D Real values
Rational functions are ratios of polynomials, so they are analytic wherever the denominator is nonzero. Points where the denominator vanishes are poles (singularities).
Euler’s formula in complex numbers is
A eiθ=cosθ+isinθeiθ=cosθ+isinθ
B eθ=cosiθeθ=cosiθ
C sinθ=icosθsinθ=icosθ
D cosθ=eθcosθ=eθ
Euler’s formula links exponentials and trigonometry. It is fundamental for writing complex numbers in polar form and for understanding rotations and periodicity in complex functions.
Which property holds for complex exponential
A ez1+z2=ez1+ez2ez1+z2=ez1+ez2
B ez=zez=z always
C ez=zˉez=zˉ
D ez1+z2=ez1ez2ez1+z2=ez1ez2
The exponential law remains valid for complex numbers. It follows from the power series definition and is crucial in simplifying expressions and solving exponential equations.
The function ex+iyex+iy can be written as
A ex(cosy+isiny)ex(cosy+isiny)
B ey(cosx+isinx)ey(cosx+isinx)
C x+iyx+iy
D cos(x+iy)cos(x+iy)
Using Euler’s formula, ex+iy=exeiy=ex(cosy+isiny)ex+iy=exeiy=ex(cosy+isiny). This shows magnitude exex and angle yy clearly.
Periodicity of eiyeiy is
A π/2π/2 in yy
B No periodicity
C 2π2π in yy
D 11 in yy
Since eiy=cosy+isinyeiy=cosy+isiny, both sine and cosine repeat every 2π2π. Hence ei(y+2π)=eiyei(y+2π)=eiy.
Principal value of \Argz\Argz is usually taken in
A [0,4π][0,4π]
B (−π,π](−π,π]
C (−∞,∞)(−∞,∞)
D [−π/2,π/2][−π/2,π/2]
The principal argument \Argz\Argz is chosen as a single representative angle, commonly in (−π,π](−π,π], to make functions like principal log single-valued on a cut domain.
A key property of complex conjugation is
A z+w‾=zwz+w=zw
B zˉ=zzˉ=z always
C 1/z‾=1/z1/z=1/z
D zw‾=zˉ wˉzw=zˉwˉ
Conjugation distributes over multiplication: zw‾=zˉ wˉzw=zˉwˉ. It also distributes over addition, helping simplify expressions and compute moduli using zzˉzzˉ.
Modulus of z=x+iyz=x+iy is
A x2+y2x2+y2
B x+yx+y
C x2+y2x2+y2
D x−yx−y
The modulus is the distance from the origin in the complex plane. For z=x+iyz=x+iy, it is ∣z∣=x2+y2∣z∣=x2+y2, matching the Pythagorean distance formula.
Which identity always holds
A ∣z∣=z+zˉ∣z∣=z+zˉ
B ∣z∣=zzˉ∣z∣=zzˉ
C ∣z∣2=zzˉ∣z∣2=zzˉ
D ∣z∣2=z+zˉ∣z∣2=z+zˉ
For z=x+iyz=x+iy, zzˉ=(x+iy)(x−iy)=x2+y2zzˉ=(x+iy)(x−iy)=x2+y2. This equals ∣z∣2∣z∣2, a very useful identity in proofs and simplifications.
Polar form of a nonzero complex number is
A z=x+yz=x+y
B z=r(cosθ+isinθ)z=r(cosθ+isinθ)
C z=r+θz=r+θ
D z=cosr+isinrz=cosr+isinr
Any nonzero complex number can be written as z=reiθ=r(cosθ+isinθ)z=reiθ=r(cosθ+isinθ), where r=∣z∣r=∣z∣ and θθ is an argument of zz.
De Moivre’s theorem states
A (cosθ+isinθ)n=cosnθ+isinnθ(cosθ+isinθ)n=cosnθ+isinnθ
B cos(θ+n)=cosθ+cosncos(θ+n)=cosθ+cosn
C ez=zeez=ze
D sinnθ=nsinθsinnθ=nsinθ
De Moivre’s theorem helps compute powers and roots of complex numbers in polar form. It is especially important for finding roots of unity and simplifying rotations.
How many distinct nnth roots does a nonzero complex number have
A n−1n−1
B 2n2n
C nn
D Infinite
A nonzero complex number has exactly nn distinct nnth roots, evenly spaced in angle around a circle in the complex plane, due to periodicity of argument.
The nnth roots of unity satisfy
A zn=0zn=0
B zn=zzn=z
C zn=zˉzn=zˉ
D zn=1zn=1
Roots of unity are solutions to zn=1zn=1. They are equally spaced points on the unit circle and form a key example for mapping and symmetry ideas.
A removable discontinuity typically can be fixed by
A Changing whole formula
B Redefining value at point
C Making function multivalued
D Taking conjugate
If the limit exists at a point but the function is undefined or wrongly defined there, assigning the limit value removes the discontinuity and makes the function continuous.
Which is a correct statement about analytic functions
A They are always continuous
B They are never continuous
C They are discontinuous at zeros
D They exist only on lines
Complex differentiability implies continuity. So if a function is analytic (holomorphic) in a region, it is automatically continuous throughout that region.
If ff is analytic, then its real and imaginary parts are
A Constant
B Discontinuous
C Harmonic
D Linear always
For analytic f=u+ivf=u+iv, both uu and vv satisfy Laplace’s equation under standard smoothness assumptions. Thus they are harmonic functions and closely linked to potential theory.
Which is an example of an entire function
A ezez
B 1/z1/z
C logzlogz
D 1/(z−1)1/(z−1)
ezez is analytic for all complex zz because its power series converges everywhere. Functions like 1/z1/z or 1/(z−1)1/(z−1) fail at their poles.
A simple pole occurs for a rational function when the denominator has
A Double zero
B No zero
C Infinite zeros
D Simple zero
A simple pole happens when the denominator has a zero of order 1 and the numerator is not zero there. Near that point, the function behaves like C/(z−a)C/(z−a).
The extended complex plane adds one extra point called
A Origin
B Infinity
C Unit point
D Branch point
The extended complex plane (Riemann sphere idea) includes an additional point at infinity. This allows treating behavior as ∣z∣→∞∣z∣→∞ similarly to behavior near a point.
The mapping w=z2w=z2 generally
A Halves argument
B Keeps argument same
C Doubles argument
D Removes modulus
In polar form z=reiθz=reiθ, squaring gives w=r2ei2θw=r2ei2θ. So the argument doubles, and the modulus becomes squared, changing angles and stretching distances.
Which mapping sends circles and lines to circles or lines
A Bilinear transform
B Squaring map
C Conjugation map
D Constant map
A bilinear (Möbius) transformation w=az+bcz+dw=cz+daz+b maps generalized circles (circles or straight lines) to generalized circles, making it central in complex mappings.
A conformal mapping mainly preserves
A Distances globally
B Areas always
C Angles locally
D Only magnitudes
Conformal mappings preserve angles and orientation at points where the derivative is nonzero. They may distort lengths and areas, but small shapes keep their angles intact.