For f(z)=z2−1z−1f(z)=z−1z2−1, what is limz→1f(z)limz→1f(z)** after simplification**
A 11
B 00
C 22
D Does not exist
Factor z2−1=(z−1)(z+1)z2−1=(z−1)(z+1). For z≠1z=1, f(z)=z+1f(z)=z+1. Hence as z→1z→1, f(z)→2f(z)→2. The singularity at 1 is removable.
If ∣f(z)−3∣≤5∣z−2∣∣f(z)−3∣≤5∣z−2∣** near z=2z=2, then limz→2f(z)limz→2f(z) equals**
A 33
B 55
C 22
D Does not exist
Since ∣z−2∣→0∣z−2∣→0, the right side 5∣z−2∣→05∣z−2∣→0. So ∣f(z)−3∣→0∣f(z)−3∣→0, meaning f(z)→3f(z)→3. This is a standard modulus-based limit estimate.
Which situation guarantees limz→af(z)=0limz→af(z)=0** using the squeeze idea**
A ∣f(z)∣≥1∣f(z)∣≥1
B ∣f(z)∣≥∣z−a∣∣f(z)∣≥∣z−a∣
C ∣f(z)∣=∣z−a∣−1∣f(z)∣=∣z−a∣−1
D ∣f(z)∣≤∣z−a∣∣f(z)∣≤∣z−a∣
If ∣f(z)∣≤∣z−a∣∣f(z)∣≤∣z−a∣ and ∣z−a∣→0∣z−a∣→0 as z→az→a, then ∣f(z)∣→0∣f(z)∣→0. Hence f(z)→0f(z)→0, so the limit exists and equals 0.
If f(z)→Lf(z)→L** as** z→az→a, then what is limz→a∣f(z)∣limz→a∣f(z)∣** (when L≠0L=0)**
A LL
B 00
C ∣L∣∣L∣
D Undefined always
Modulus is continuous, so limits pass through it: lim∣f(z)∣=∣limf(z)∣lim∣f(z)∣=∣limf(z)∣. Therefore if f(z)→Lf(z)→L, then ∣f(z)∣→∣L∣∣f(z)∣→∣L∣. This is a key limit theorem.
If ff and gg both have limits at aa, then limz→a(f+g)limz→a(f+g) equals
A Quotient of limits
B Sum of limits
C Product of limits
D Difference of limits
Limit laws hold in CC the same way as real limits. If limf=Llimf=L and limg=Mlimg=M, then lim(f+g)=L+Mlim(f+g)=L+M, provided both limits exist.
For limz→∞3z2+1z2−5limz→∞z2−53z2+1, the limit equals
A 00
B ∞∞
C 33
D −3−3
Divide numerator and denominator by z2z2. As ∣z∣→∞∣z∣→∞, 1/z2→01/z2→0. The ratio approaches 3+01+0=31+03+0=3. Same as real rational limits.
For limz→∞2z+7z2+1limz→∞z2+12z+7, the limit equals
A 11
B 22
C ∞∞
D 00
Degree of denominator (2) exceeds numerator (1). Divide by z2z2: 2/z+7/z21+1/z2→01+1/z22/z+7/z2→0. Hence the limit at infinity is 0.
If ff is continuous at aa** and gg is continuous at f(a)f(a), then g∘fg∘f is
A Discontinuous at aa
B Multivalued at aa
C Undefined at aa
D Continuous at aa
Composition of continuous functions is continuous. If f(z)→f(a)f(z)→f(a) as z→az→a and g(w)→g(f(a))g(w)→g(f(a)) as w→f(a)w→f(a), then g(f(z))→g(f(a))g(f(z))→g(f(a)).
Which function is continuous where defined but has a discontinuity at its pole
A z3z3
B ezez
C 1z−az−a1
D sinzsinz
1/(z−a)1/(z−a) is continuous for all z≠az=a. At z=az=a, it is not defined and blows up in magnitude, so continuity fails there due to a pole.
If f(z)=u+ivf(z)=u+iv, which pair is correct for u,vu,v being real-valued
A Depend on x,yx,y
B Depend only on zz
C Always constants
D Always equal
With z=x+iyz=x+iy, we write f(z)=u(x,y)+iv(x,y)f(z)=u(x,y)+iv(x,y). Both uu and vv are real functions of two real variables and describe real/imaginary parts.
A function is single-valued if each input zz gives
A Two output values
B Infinite outputs
C One output value
D No output
Single-valued means for each zz in the domain, the function assigns exactly one complex number. Multivalued examples like logzlogz need branches to choose one value.
The principal branch of logzlogz commonly uses
A \Argz∈(0,π)\Argz∈(0,π)
B \Argz∈(−π,π]\Argz∈(−π,π]
C \Argz∈(−2π,2π)\Argz∈(−2π,2π)
D \Argz∈(0,4π)\Argz∈(0,4π)
The principal argument is usually chosen in (−π,π](−π,π]. Using this choice gives the principal logarithm \Logz=ln∣z∣+i\Argz\Logz=ln∣z∣+i\Argz, making it single-valued on a cut plane.
Which mapping sends the unit circle ∣z∣=1∣z∣=1 to itself
A w=eiθzw=eiθz
B w=2zw=2z
C w=z+1w=z+1
D w=z2+1w=z2+1
Multiplying by eiθeiθ preserves modulus: ∣w∣=∣eiθ∣∣z∣=∣z∣∣w∣=∣eiθ∣∣z∣=∣z∣. So points with ∣z∣=1∣z∣=1 remain on ∣w∣=1∣w∣=1, just rotated.
Under translation w=z+aw=z+a, which property is preserved
A Distance from origin
B Argument of points
C Modulus becomes squared
D Distances between points
Translation shifts every point by the same vector, so differences z1−z2z1−z2 remain unchanged. Hence distances between points are preserved, though distance from origin generally changes.
Under scaling w=czw=cz** with c>0c>0, what happens to angles at origin**
A Angles doubled
B Angles become zero
C Angles unchanged
D Angles random
If c>0c>0 is real, w=czw=cz multiplies modulus by cc and keeps argument the same. So rays from the origin keep their angles, though lengths are scaled.
For inversion w=1/zw=1/z, a circle through origin maps to a
A Line not through origin
B Circle through origin
C Fixed circle only
D Point at origin
Inversion is a special Möbius map. It sends circles/lines to circles/lines. A circle passing through 00 maps to a straight line (not passing through 00).
If ff is differentiable at aa, then it must be
A Discontinuous at aa
B Multivalued at aa
C Continuous at aa
D Undefined at aa
Complex differentiability implies continuity. If the derivative limit exists at aa, then f(z)→f(a)f(z)→f(a) as z→az→a. So differentiability is stronger than continuity.
A quick reason f(z)=∣z∣2f(z)=∣z∣2 is not analytic is
A Not continuous
B CR fails generally
C Not defined at 0
D Multivalued outputs
f(z)=∣z∣2=x2+y2f(z)=∣z∣2=x2+y2 is real-valued. Writing u=x2+y2, v=0u=x2+y2, v=0, CR gives ux=2xux=2x and vy=0vy=0, so equality fails except at special points.
If f(z)=zzˉf(z)=zzˉ, then ff is
A Analytic everywhere
B Entire function
C Constant function
D Real-valued function
Since zzˉ=∣z∣2≥0zzˉ=∣z∣2≥0, the output is always real. Such functions are typically not analytic because CR conditions force strong restrictions not met by x2+y2x2+y2.
For f(z)=ezf(z)=ez, what is f′(z)f′(z)
A zezzez
B ezˉezˉ
C ezez
D 1/ez1/ez
The derivative of ezez equals itself, because its power series differentiates term-by-term back to the same series. This holds for complex zz just like real xx.
The complex chain rule says (f∘g)′(z)(f∘g)′(z) equals
A f′(g(z))g′(z)f′(g(z))g′(z)
B f′(z)+g′(z)f′(z)+g′(z)
C f′(z)g(z)f′(z)g(z)
D f(z)g′(z)f(z)g′(z)
If gg is differentiable at zz and ff differentiable at g(z)g(z), then (f∘g)′(z)=f′(g(z))g′(z)(f∘g)′(z)=f′(g(z))g′(z). Same rule as real calculus.
If f(z)=z3f(z)=z3** and g(z)=z2g(z)=z2, then (f∘g)(z)(f∘g)(z) equals
A z5z5
B z3+z2z3+z2
C z6z6
D z9z9
(f∘g)(z)=f(g(z))=(z2)3=z6(f∘g)(z)=f(g(z))=(z2)3=z6. Composition means apply gg first, then ff. This also helps practice mapping behavior.
For f(z)=z2f(z)=z2, the derivative at z=1+iz=1+i is
A 2+2i2+2i
B 1+i1+i
C 22
D 2i2i
Since f′(z)=2zf′(z)=2z, evaluate at z=1+iz=1+i: f′(1+i)=2(1+i)=2+2if′(1+i)=2(1+i)=2+2i. Complex derivatives are evaluated the same way as real ones.
Cauchy–Riemann equations require partial derivatives of u,vu,v to be
A Always zero
B Always equal
C Existing at point
D Not existing
To apply CR, ux,uy,vx,vyux,uy,vx,vy must exist at the point. For sufficiency of analyticity, these partial derivatives are typically required to be continuous in a neighborhood.
For f(z)=u+ivf(z)=u+iv, if ux=vyux=vy and uy=−vxuy=−vx, then
A ff multivalued
B ff discontinuous
C ff constant
D ff differentiable
If CR equations hold and the partial derivatives are continuous near the point, then ff is complex differentiable there. This is a standard sufficient condition for analyticity.
If ff is analytic, then uu satisfies
A uxx−uyy=0uxx−uyy=0
B uxx+uyy=0uxx+uyy=0
C ux+uy=0ux+uy=0
D uxy=1uxy=1
Real and imaginary parts of an analytic function are harmonic. That means they satisfy Laplace’s equation: uxx+uyy=0uxx+uyy=0 and vxx+vyy=0vxx+vyy=0 under usual smoothness assumptions.
If a nonzero analytic function has infinitely many zeros accumulating inside a region, then
A Zeros become poles
B Function becomes multivalued
C Function is identically zero
D Function loses continuity
Zeros of a nonzero analytic function are isolated. If zeros accumulate at a point inside the domain, the identity theorem implies the function must be zero everywhere on that region.
The equation ez=1ez=1** has solutions**
A 2πik2πik
B πikπik
C 2πk2πk
D ikik only
ez=1ez=1 means ex+iy=ex(cosy+isiny)=1ex+iy=ex(cosy+isiny)=1. This requires x=0x=0 and y=2πky=2πk. So z=2πikz=2πik, integers kk.
The equation ez=−1ez=−1** has solutions**
A 2kπi2kπi
B kπkπ
C (2k+1)πi(2k+1)πi
D kiki
−1=eiπ−1=eiπ and also ei(π+2πk)ei(π+2πk). So ez=−1ez=−1 requires x=0x=0 and y=(2k+1)πy=(2k+1)π. Hence z=(2k+1)πiz=(2k+1)πi.
If z=reiθz=reiθ, then principal value of \Logz\Logz is
A r+iθr+iθ
B lnθ+irlnθ+ir
C reiθreiθ
D lnr+iΘlnr+iΘ
The principal logarithm is \Logz=ln∣z∣+i\Argz\Logz=ln∣z∣+i\Argz. Writing z=reiθz=reiθ, it becomes lnr+iΘlnr+iΘ, where Θ=\ArgzΘ=\Argz in the chosen principal range.
A branch point is linked to functions like
A z2z2
B logzlogz
C z+1z+1
D ezez
logzlogz changes by 2πi2πi when circling the origin, so the value depends on the path. This behavior signals a branch point, typically at z=0z=0.
For roots of unity, the solutions of zn=1zn=1 lie on
A Real axis
B Imaginary axis
C Unit circle
D Parabola curve
If zn=1zn=1, then ∣z∣n=1∣z∣n=1 so ∣z∣=1∣z∣=1. Thus all nth roots of unity lie on the unit circle, equally spaced in angle.
The arguments of nth roots of unity are
A 2πk/n2πk/n
B πk/nπk/n
C k/nk/n
D 2k/n2k/n
Write 1=ei2πk1=ei2πk. Solving zn=ei2πkzn=ei2πk gives z=ei2πk/nz=ei2πk/n. Hence the roots are at angles 2πk/n2πk/n, k=0,…,n−1k=0,…,n−1.
A Möbius transformation is conformal where its derivative is
A Zero
B Infinite only
C Nonzero
D Undefined always
Möbius maps are analytic where defined. They preserve angles at points where the derivative is not zero. At points where derivative vanishes or map is undefined, conformality can fail.
If w=az+bcz+dw=cz+daz+b, then the condition to avoid constant mapping is
A a=b=c=da=b=c=d
B c=0c=0 always
C b=d=0b=d=0
D ad−bc≠0ad−bc=0
For a Möbius transformation, ad−bc≠0ad−bc=0 ensures it is a genuine one-to-one transformation (except at its pole). If ad−bc=0ad−bc=0, it can collapse to a constant map.
For analytic ff, the existence of f′(a)f′(a) means the difference quotient limit is
A Same on x-axis
B Same on y-axis
C Same all directions
D Same on circle only
In complex differentiability, z→az→a can occur from any direction. The derivative exists only if the limit of f(z)−f(a)z−az−af(z)−f(a) is identical for every approach path.
Which is a correct derivative rule in complex calculus
A (fg)′=f′g′(fg)′=f′g′
B (fg)′=f′g+fg′(fg)′=f′g+fg′
C (f+g)′=fg(f+g)′=fg
D (f/g)′=f′/g′(f/g)′=f′/g′
Product rule holds in complex calculus exactly as in real calculus. If ff and gg are differentiable at zz, then (fg)′(z)=f′(z)g(z)+f(z)g′(z)(fg)′(z)=f′(z)g(z)+f(z)g′(z).
If f(z)=sinzf(z)=sinz, which statement is correct
A Has a pole
B Multivalued always
C Entire function
D Undefined on real
sinzsinz has a power series ∑(−1)nz2n+1/(2n+1)!∑(−1)nz2n+1/(2n+1)! that converges for all zz. Therefore sinzsinz is analytic everywhere, so it is entire.
If f(z)=coszf(z)=cosz, which statement is correct
A Entire function
B Pole at zero
C Branch cut needed
D Not continuous
coszcosz is defined by a power series that converges for every complex zz. Hence it is analytic on all of CC, making it an entire function.
A “removable” singularity becomes analytic if we
A Add branch cut
B Take conjugate
C Square the function
D Redefine value
If limz→af(z)=Llimz→af(z)=L exists and is finite, then defining f(a)=Lf(a)=L removes the singularity. The new function becomes continuous and often analytic at aa.
A simple example of a pole of order 2 is
A 1/(z−a)1/(z−a)
B log(z−a)log(z−a)
C 1/(z−a)21/(z−a)2
D ez−aez−a
1/(z−a)21/(z−a)2 has a second-order pole at z=az=a because the denominator has a zero of order 2. The function magnitude blows up like 1/∣z−a∣21/∣z−a∣2.
In Laurent series, the “principal part” means terms with
A Negative powers
B Positive powers
C Only constant term
D Only even powers
A Laurent series around aa includes both positive and negative powers of (z−a)(z−a). The principal part is the sum of negative-power terms and describes the singular behavior.
A pole corresponds to Laurent principal part having
A Infinite negative terms
B Finite negative terms
C No negative terms
D Only constant term
At a pole of order mm, the Laurent series has negative powers down to (z−a)−m(z−a)−m and stops. Only essential singularities have infinitely many negative-power terms.
An essential singularity is indicated when Laurent series has
A Exactly one negative
B No negative powers
C Infinite negative powers
D Only positive powers
If the principal part has infinitely many negative-power terms, the point is an essential singularity. This creates extremely irregular behavior near the point compared with poles or removable singularities.
If a function is analytic on and inside a simple closed contour, then ∮f(z) dz∮f(z)dz equals
A 11
B 2πi2πi
C Depends on path
D 00
Cauchy’s integral theorem states that if ff is analytic throughout a simply connected region containing the contour and its interior, then the integral around that closed contour is zero.
Residue at z=az=a is the coefficient of
A (z−a)0(z−a)0
B (z−a)1(z−a)1
C (z−a)−1(z−a)−1
D (z−a)−2(z−a)−2
In the Laurent series of ff about aa, the residue is the coefficient of (z−a)−1(z−a)−1. It is used in the residue theorem to compute contour integrals.
If z=c+iyz=c+iy, then w=ec(cosy+isiny)w=ec(cosy+isiny). The modulus is constant ecec while yy changes angle, producing a circle centered at the origin.
If z=x+ikz=x+ik, then w=ex(cosk+isink)w=ex(cosk+isink). The angle is fixed by kk while the modulus varies with exex, forming a ray from the origin.
In polar form, multiplication of two complex numbers causes their arguments to
A Subtract always
B Become equal
C Become zero
D Add together
If z1=r1eiθ1z1=r1eiθ1 and z2=r2eiθ2z2=r2eiθ2, then z1z2=r1r2ei(θ1+θ2)z1z2=r1r2ei(θ1+θ2). So moduli multiply and arguments add.
A basic reason logzlogz is not entire is that it
A Is polynomial
B Has no zeros
C Needs branch cut
D Is constant
logzlogz is multivalued because the argument can change by 2πk2πk. Any single-valued branch needs a cut and excludes a region, so logzlogz cannot be analytic on all CC.