While checking limz→af(z)limz→af(z), which approach best confirms path independence
A Compare two different paths
B Check only real axis
C Check only imaginary axis
D Use just continuity
A complex limit must be the same for every path toward aa. Testing two different paths is a quick way to detect failure; if values differ, the limit cannot exist.
If limz→af(z)=Llimz→af(z)=L, which inequality expresses the epsilon–delta idea
A ∣f(z)−L∣<ε∣f(z)−L∣<ε
B ∣f(z)+L∣<ε∣f(z)+L∣<ε
C ∣z+a∣<ε∣z+a∣<ε
D ∣f(z)∣>ε∣f(z)∣>ε
The definition says: for every ε>0ε>0, there is δ>0δ>0 such that if 0<∣z−a∣<δ0<∣z−a∣<δ, then ∣f(z)−L∣<ε∣f(z)−L∣<ε. This matches complex limits.
Which statement is correct for a constant complex function f(z)=cf(z)=c** near any point**
A Limit depends on path
B Limit equals cc
C Limit is infinity
D Limit never exists
If f(z)=cf(z)=c for all zz, then as z→az→a the function value stays cc. Therefore the limit exists and equals cc, independent of approach direction.
When studying limz→∞f(z)limz→∞f(z), the point “∞∞” is treated as
A A single added point
B A real number
C An imaginary number
D A branch cut
In the extended complex plane, infinity is treated as one extra point. This helps describe behavior as ∣z∣→∞∣z∣→∞ similarly to limits as z→az→a.
If ∣f(z)∣≤∣z−a∣2∣f(z)∣≤∣z−a∣2** near aa, then limz→af(z)limz→af(z) is**
A 11
B Does not exist
C Infinity
D 00
Since ∣z−a∣2→0∣z−a∣2→0 as z→az→a, the squeeze idea gives ∣f(z)∣→0∣f(z)∣→0. Hence f(z)→0f(z)→0, so the limit exists and equals 00.
Which function is continuous at every complex number
A z5−3zz5−3z
B 1/(z−2)1/(z−2)
C logzlogz
D 1/zˉ1/zˉ
Polynomials are continuous everywhere in CC. The other options fail at some points due to division by zero, multivalued behavior, or lack of analyticity.
For f(z)=z+1z−1f(z)=z−1z+1, the function is continuous at z=2z=2 because
A Denominator nonzero
B Numerator nonzero
C Argument equals zero
D Modulus equals one
A rational function is continuous wherever its denominator is not zero. At z=2z=2, the denominator 2−1=1≠02−1=1=0, so the function is defined and continuous there.
Which expression represents the real part of z=x+iyz=x+iy
A yy
B xx
C ∣z∣∣z∣
D argzargz
A complex number is written z=x+iyz=x+iy, where xx is the real part and yy is the imaginary part. This basic split is used in u(x,y)+iv(x,y)u(x,y)+iv(x,y) forms.
If f(z)=u(x,y)+iv(x,y)f(z)=u(x,y)+iv(x,y), then u(x,y)u(x,y) is
A Real-valued part
B Imaginary-valued part
C Modulus function
D Argument function
The function u(x,y)u(x,y) gives the real component of f(z)f(z), while v(x,y)v(x,y) gives the imaginary component. Both are real-valued functions of two variables.
Which function commonly needs a branch choice to be single-valued
A z3z3
B z+2z+2
C zz
D ezez
zz has two values for each nonzero zz, so it is multivalued. A branch cut and a branch choice restrict it to one value on a chosen domain.
The mapping w=z−3w=z−3 moves every point
A Right by 3
B Rotates by 3
C Inverts about origin
D Left by 3
Subtracting 3 shifts the real part by −3−3, moving points left along the real axis by 3 units. Shape and angles remain unchanged because it’s a translation.
The mapping w=2zw=2z does what to distances from origin
A Doubles all distances
B Halves all distances
C Keeps distances same
D Makes distance zero
If w=2zw=2z, then ∣w∣=2∣z∣∣w∣=2∣z∣. So every point is scaled away from the origin by factor 2, with angles preserved because the multiplier is real positive.
If w=eiπ/2zw=eiπ/2z, then the mapping is a
A 90° rotation
B 180° rotation
C Reflection in x-axis
D Translation upward
eiπ/2=ieiπ/2=i. Multiplying by ii rotates every point counterclockwise by 90∘90∘, keeping magnitudes unchanged and adding π/2π/2 to arguments.
Under mapping w=z2w=z2, the argument changes from θθ to
A 2θ2θ
B θ/2θ/2
C θ+1θ+1
D −θ−θ
Writing z=reiθz=reiθ, squaring gives w=r2ei2θw=r2ei2θ. So the angle doubles while the modulus squares, which affects how rays and regions map.
What happens to a point near infinity under w=1/zw=1/z** (excluding z=0z=0)**
A Moves near 0
B Moves to infinity
C Stays unchanged
D Becomes multivalued
Since ∣w∣=1/∣z∣∣w∣=1/∣z∣, if ∣z∣∣z∣ is very large then ∣w∣∣w∣ is very small. So points far away map close to the origin under inversion.
Which is the correct difference quotient for complex derivative at aa
A f(a)−f(z)a−za−zf(a)−f(z)
B f(z)+f(a)z−az−af(z)+f(a)
C f(z)−az−f(a)z−f(a)f(z)−a
D f(z)−f(a)z−az−af(z)−f(a)
The complex derivative is defined exactly like the real derivative but with complex zz. The limit as z→az→a must exist and be independent of direction.
If f(z)=z2+1f(z)=z2+1, then f′(z)f′(z) is
A 2z2z
B zz
C z2z2
D 22
Using the power rule for complex polynomials, ddz(z2)=2zdzd(z2)=2z and constant derivative is 0. So f′(z)=2zf′(z)=2z, valid for all complex zz.
If f(z)=az+bf(z)=az+b** with constants a,ba,b, then the derivative is**
A bb
B a+ba+b
C aa
D abab
Linear functions behave the same in complex calculus. The slope aa is constant, and the derivative of the constant term bb is 0, so f′(z)=af′(z)=a.
Cauchy–Riemann equations are used mainly to test
A Analyticity at a point
B Continuity at infinity
C Modulus of zz
D Zeros of polynomials
For f=u+ivf=u+iv, the CR equations relate partial derivatives of uu and vv. Under usual smoothness, satisfying CR implies complex differentiability, hence analyticity near that point.
If ux=vyux=vy and uy=−vxuy=−vx, then
A CR satisfied
B Function constant
C Limit fails
D Mapping not defined
These are exactly the Cartesian Cauchy–Riemann equations. They are necessary for differentiability and, with continuity of partial derivatives in a neighborhood, become sufficient for analyticity.
For f(z)=x+iyf(z)=x+iy, the function equals
A zˉzˉ
B ∣z∣∣z∣
C zz
D argzargz
By definition, z=x+iyz=x+iy. So the function f(z)=x+iyf(z)=x+iy is simply the identity function f(z)=zf(z)=z, which is analytic everywhere with derivative 1.
For f(z)=zˉ=x−iyf(z)=zˉ=x−iy, which statement is true
A CR fails everywhere
B CR holds everywhere
C Derivative equals 1
D Entire function
Here u=xu=x, v=−yv=−y. Then ux=1ux=1 while vy=−1vy=−1, so ux≠vyux=vy. Hence CR fails at every point and ff is nowhere analytic.
If a function is analytic in a region, it must be
A Continuous in region
B Discontinuous in region
C Only real-valued
D Only bounded
Complex differentiability is a strong condition and implies continuity. Therefore any analytic (holomorphic) function is automatically continuous throughout the region where it is analytic.
An “entire” function is analytic on
A All complex numbers
B Only unit circle
C Only real axis
D Only outside disk
Entire means analytic everywhere in CC. Standard examples include polynomials and ezez. Functions with poles or branch issues cannot be entire.
Which function is not analytic at z=0z=0
A 1/z1/z
B z2z2
C ezez
D sinzsinz
1/z1/z has a pole at z=0z=0, so it is not defined and not analytic there. The other listed functions are analytic everywhere and are entire.
A “pole” of a function is a type of
A Singularity
B Zero value
C Constant region
D Branch cut
A pole is a point where a function blows up to infinity in magnitude, like 1/(z−a)1/(z−a) at z=az=a. It is one common type of singularity in complex analysis.
A removable singularity is characterized by
A Finite limit exists
B Limit is infinite
C No limit along paths
D Always multivalued
If ff is not defined at aa but limz→af(z)limz→af(z) exists and is finite, the singularity is removable. Defining f(a)f(a) as that limit fixes it.
Euler’s formula directly connects
A Exponential and trig
B Polynomials and logs
C Limits and sequences
D Modulus and zeros
Euler’s formula eiθ=cosθ+isinθeiθ=cosθ+isinθ links complex exponentials to trigonometric functions. It explains rotation, periodicity, and polar representation compactly.
For z=x+iyz=x+iy, the value of ezez equals
A ey(cosx+isinx)ey(cosx+isinx)
B cosx+isinycosx+isiny
C x+ieyx+iey
D ex(cosy+isiny)ex(cosy+isiny)
Using ex+iy=exeiyex+iy=exeiy and Euler’s formula for eiyeiy, we get ex(cosy+isiny)ex(cosy+isiny). This shows magnitude exex and angle yy.
Which statement about eiyeiy is correct
A Period 2π2π
B Period ππ
C Not periodic
D Period 11
Since eiy=cosy+isinyeiy=cosy+isiny, and both cosine and sine repeat every 2π2π, we have ei(y+2π)=eiyei(y+2π)=eiy, giving period 2π2π.
The principal logarithm \Logz\Logz uses
A Principal argument
B Any argument
C Only real argument
D Only imaginary part
The principal logarithm \Logz\Logz is a single-valued choice of logzlogz using the principal argument \Argz\Argz. It typically requires a branch cut to stay consistent.
Which identity is always true for complex conjugates
A z+w‾=zˉ+wˉz+w=zˉ+wˉ
B z+w‾=zwz+w=zw
C zw‾=z+wzw=z+w
D z‾=argzz=argz
Conjugation distributes over addition: z+w‾=zˉ+wˉz+w=zˉ+wˉ. This property helps simplify expressions, prove modulus relations, and manipulate complex equations cleanly.
If ∣z∣=r∣z∣=r, then zzˉzzˉ equals
A rr
B r2r2
C 1/r1/r
D −r2−r2
For any complex number, zzˉ=∣z∣2zzˉ=∣z∣2. If ∣z∣=r∣z∣=r, then zzˉ=r2zzˉ=r2. This is a key identity for converting to real quantities.
In polar form, a nonzero complex number is written as
A r+iθr+iθ
B x+iyx+iy only
C rθrθ
D reiθreiθ
For z≠0z=0, polar form is z=reiθz=reiθ, where r=∣z∣r=∣z∣ and θθ is an argument. It simplifies multiplication, powers, and roots using angles.
De Moivre’s theorem is mainly used to compute
A Powers and roots
B Only sums
C Only limits
D Only derivatives
De Moivre gives (cosθ+isinθ)n=cosnθ+isinnθ(cosθ+isinθ)n=cosnθ+isinnθ. It makes powers easy and leads to the standard method for finding nth roots in polar form.
How many solutions does z4=1z4=1 have in complex numbers
A Four solutions
B Two solutions
C One solution
D Infinite solutions
The equation zn=1zn=1 has exactly nn distinct complex solutions, equally spaced on the unit circle. So z4=1z4=1 has 4 roots at angles 0,π/2,π,3π/20,π/2,π,3π/2.
The modulus function ∣z∣∣z∣ is
A Continuous everywhere
B Discontinuous everywhere
C Analytic everywhere
D Multivalued always
∣z∣∣z∣ depends continuously on (x,y)(x,y) via x2+y2x2+y2, so it is continuous on CC. However, it is not analytic because CR conditions fail except in trivial cases.
A conformal map preserves
A Angles locally
B Areas always
C Distances always
D Only real parts
Conformal mappings preserve angles and their orientation at points where the derivative is nonzero. They can stretch lengths and areas, but small shapes keep the same angles.
A bilinear transformation has the form
A az2+bz+caz2+bz+c
B eaz+beaz+b
C zˉ+azˉ+a
D az+bcz+dcz+daz+b
Möbius (bilinear) transformations are az+bcz+dcz+daz+b with ad−bc≠0ad−bc=0. They map lines/circles to lines/circles and are central in complex mapping theory.
A key geometric fact about Möbius maps is
A Circles map to circles
B Areas stay same
C Distances stay same
D All points fixed
Möbius transformations map generalized circles (circles or straight lines) into generalized circles. This property is widely used to find images of lines, circles, and regions under transformations.
If ff is analytic, then uu and vv typically satisfy
A Logistic equation
B Wave equation
C Laplace equation
D Newton equation
For analytic f=u+ivf=u+iv, both uu and vv are harmonic under standard smoothness conditions, meaning they satisfy Laplace’s equation. This connects complex analysis with potential theory.
“Harmonic conjugate” refers to a function vv such that
A u+ivu+iv analytic
B u=vu=v always
C v=uˉv=uˉ
D vv multivalued
If uu is harmonic and there exists vv making f=u+ivf=u+iv analytic, then vv is called a harmonic conjugate of uu. CR equations link their partial derivatives.
In basic terms, a contour in complex analysis is a
A Directed curve
B Single point only
C Flat region only
D Constant mapping
A contour is a curve in the complex plane with a direction. It is used in complex integration, where the integral depends on the path, orientation, and function behavior along the curve.
Cauchy integral theorem broadly states that for analytic functions, contour integrals over closed loops are
A Zero
B Infinite
C Always real
D Always imaginary
If a function is analytic throughout a simply connected region, the integral around any closed contour in that region is 0. This is a foundational result leading to many powerful formulas.
A “residue” is mainly used to compute
A Contour integrals
B Polynomial roots
C Function continuity
D Argument values
Residues capture the coefficient of 1/(z−a)1/(z−a) in a Laurent expansion. The residue theorem uses these residues to evaluate many complex contour integrals efficiently.
A Laurent series is used near a point to represent
A Functions with singularity
B Only polynomials
C Only continuous maps
D Only constant values
Laurent series extends power series by allowing negative powers of (z−a)(z−a). This makes it suitable for describing functions near isolated singularities, such as poles and essential singularities.
An essential singularity is a point where the Laurent series has
A Infinitely many negative powers
B No negative powers
C Only one negative power
D No series exists
At an essential singularity, the principal part of the Laurent series contains infinitely many negative-power terms. This causes highly irregular behavior near the singular point.
The term “analytic continuation” means
A Extend analytic function
B Differentiate repeatedly
C Replace by conjugate
D Make branch cut
Analytic continuation extends an analytic function from one region to a larger region while keeping values consistent on the overlap. It is common with functions defined by series or branches.
Which statement about zeros of a nonzero analytic function is true
A Zeros form a region
B Zeros are everywhere
C Zeros are isolated
D No zeros possible
A nonzero analytic function cannot have an accumulation of zeros inside its domain. If zeros cluster in a region, the function must be identically zero, so usual zeros are isolated points.
A basic use of the substitution w=1/zw=1/z is to convert
A Infinity to zero
B Zero to infinity
C Circle to line
D Line to circle
Setting w=1/zw=1/z turns behavior as ∣z∣→∞∣z∣→∞ into behavior as w→0w→0. This is a standard technique for evaluating limits at infinity in complex analysis.