Aktosun, Spring 2023,
Math 5322, Supplementary Problems 2
- Determine where the following functions are analytic:
- ez
solution:
everywhere
- sinz
solution:
everywhere
- cosz
solution:
everywhere
- tanz
solution:
everywhere except z=π/2+nπ, n=0,±1,±2,....
- cotz
solution:
everywhere except z=nπ, n=0,±1,±
2,....
- secz
solution:
everywhere except z=π/2+nπ, n=0,±1,±
2,....
- cscz
solution:
everywhere except z=nπ, n=0,±1,±
2,....
- sinhz
solution:
everywhere
- coshz
solution:
everywhere
- tanhz
solution:
everywhere except z=(π/2+nπ)i, n=0,±1,±
2,....
- sechz
solution:
everywhere except z=(π/2+nπ)i, n=0,±1,±
2,....
- cothz
solution:
everywhere except z=nπi, n=0,±1,±
2,....
- cschz
solution:
everywhere except z=nπi, n=0,±1,
±
2,....
- zn, n=0,1,2,....
solution:
everywhere
- z-n, n=1,2,3,....
solution:
everywhere except at z=0
- principal branch of
z1/n, n=2,3,4,....
solution:
everywhere except at z=0
- Logz
solution:
everywhere except on the nonpositive real axis.
- rational function
solution:
everywhere except at zeros of the denominator
- meromorphic function
solution:
everywhere except at the poles
- Argz
solution:
nowhere
- Express the following in terms of complex logarithms:
- sin-1z
solution:
-ilog(iz+(1-z2)1/2)
- cos-1z
solution:
-ilog(z+i(1-z2)1/2)
- tan-1z
solution:
-(i/2)log((1+iz)/(1-iz))
- cot-1z
solution:
-(i/2)log((z+i)/(z-i))
- sec-1z
solution:
-ilog(1/z+(1/z2-1)1/2)
- csc-1z
solution:
-ilog(i/z+(1-1/z2)1/2)
- sinh-1z
solution:
log(z+(1+z2)1/2)
- cosh-1z
solution:
log(z+(z2-1)1/2)
- tanh-1z
solution:
(1/2)log((1+z)/(1-z))
- coth-1z
solution:
(1/2)log((z+1)/(z-1))
- sech-1z
solution:
log(1/z+(1/z2-1)1/2)
- csch-1z
solution:
log(1/z+(1+1/z2)1/2)
- Evaluate the following:
- log(4)
solution:
ln(4)+2nπi, n=0,±1,±2,....
- Log(4)
solution:
ln(4)
- log(-1)
solution:
(2n+1)πi, n=0,±1,±2,....
- Log(-1)
solution:
πi
- log(i)
solution:
(1/2+2n)πi, n=0,±1,±2,....
- Log(i)
solution:
πi/2
- log(2-3i)
solution:
(1/2)ln(13)+(2nπ-a)i,
where a=tan-1(3/2) and
n=0,±1,±2,....
- Log(2-3i)
solution:
(1/2)ln(13)-αi,
where α=tan-1(3/2)
- log((1±i)/√(2))
solution:
(±1/4+2n)πi, n=0,±1,±2,....
- ii
solution:
e(2n-1/2)π,
n=0,±1,±2,....
- (-2)√(2)
solution:
e(ln2+(2n+1)πi)√(2),
n=0,±1,±2,....
- 2i
solution:
eiln2+2nπ,
n=0,±1,±2,....
- 1-i
solution:
e2nπ,
n=0,±1,±2,....
- ((1±i)/√(2))1+i
solution:
e(1+i)(±1/4+2n)πi,
n=0,±1,±2,....
- (3-4i)1+i
solution:
e(1+i)(ln5+(2nπ-a)i,
where a=tan-1(4/3) and
n=0,±1,±2,....
- log1+i(1-i)
solution:
log(1-i)/log(1+i)
or equivalently
[(ln2)/2-πi/4+2nπi]/[(ln2)/2+πi/4+2mπi],
n,m=0,±1,±2,....
- 1√(2)
solution:
e2√(2)nπi,
n=0,±1,±2,....
- e1/2
solution:
√(e)
- (e+i0)1/2
solution:
±√(e)
- ei
solution:
ei or cos(1)+isin(1)
- (e+i0)i
solution:
e(i+2nπ),
n=0,±1,±2,....
- sin-1(1/2)
solution:
(2n+1/6)π or
(2n+5/6)π,
n=0,±1,±2,....
- cos-1(1/2)
solution:
(2n+1/3)π
or (2n-1/3)π,
n=0,±1,±2,....
- cos-1(2)
solution:
-iln(2±√(3))+2nπ,
n=0,±1,±2,....
- sin-1(i)
solution:
-iln(-1+√(2))+2nπ
or -iln(1+√(2))+(2n+1)π,
n=0,±1,±2,....
- tanh-1(1+2i)
solution:
(1/4)ln(2)+(n+3/8)πi,
n=0,±1,±2,....
- cosh-1(2i)
solution:
ln(√(5)+2)+(2n+1/2)πi
or ln(√(5)-2)+(2n-1/2)πi,
n=0,±1,±2,....
- tanh-1(1-i)
solution:
(1/4)ln(5)+(nπ-π/2+a/2)i,
where a=tan-1(2) and
n=0,±1,±2,....
- True or false:
- log(z1z2)=log(z1)+log(z2)
solution:
true for any nonzero z1, z2
- log(z)+log(z)=2log(z)
solution:
not true in general
- log(z)+log(z)+log(z)=3log(z)
solution:
not true in general
- log(z-1)=-log(z)
solution:
true for any nonzero z
- log(zn)=nlog(z)
with n=2,3,....
solution:
not true in general
- log(z1/n)=(1/n)log(z)
with n=2,3,....
solution:
true for any nonzero z
- log(z1/z2)=log(z1)-log(z2)
solution:
true for any nonzero z1, z2
- log(1/z)=-log(z)
solution:
true for any nonzero z
- Log(z1z2)=Log(z1)+Log(z2)
solution:
not true in general, e.g. let z1=z2=-i
- Log(zn)=nLog(z)
solution:
true in general only for nonzero z and n=1
- Log(z)+Log(z)=2Log(z)
solution:
true for any nonzero z
- Log(z1/z2)=Log(z1)-Log(z2)
solution:
not true in general, e.g. let z1=1 and z2=-1
- Log(1/z)=-Log(z)
solution:
not true in general, e.g. let z=-1
- Characterize the zeros and singularities of the following functions:
- (3z2+z)/(z2+z)
solution:
simple zero at z=-1/3, simple pole at z=-1, removable singularity at z=0
- cos2(2z+3i)
solution:
double zeros at z=-3i/2+(n/2+1/4)π, n=0,±1,±2,....
- sech2z
solution:
double poles at z=(n+1/2)πi,
n=0,±1,±2,....
- z/(z4+4)
solution:
simple zero at z=0, simple poles at z=±√(2)eπi/4 and
z=±√(2)e-πi/4
- tan4z
solution:
fourth-order zeros at z=nπ
and fourth-order poles at z=(n+1/2)π,
n=0,±1,±2,....
- tan(z2)
solution:
simple zeros at z=±√(nπ) and
simple poles at z=±√[(n+1/2)π],
n=0,±1,±2,....
- ez/(z2+1)
solution:
simple poles at z=±i
- (sinz)/[z(z2+1)]
solution:
simple poles at z=±i, removable singularity at z=0,
simple zeros at z=nπ,
n=±1,±2,±3,....
Tuncay Aktosun
aktosun@uta.edu
Last modified: November 28, 2022