Chapter 2: Special Functions

Question 1

exp(z)

as a power series defined everywhere on C

As a limit

And more importantly….

Answer 1

For z= x+iy

Exp(z) = sum from 0 to ∞ ( z^n / (n!))

Or
= limit as n tends to ∞ ( 1 + (z/n))^n

And so ( 1+ (x/n))^n tends to e^x as n tends to ∞
OR

Exp (z) = exp(x) (cosy * i siny)

Question 2

Theorem 2.2

Properties of the exponential function

Answer 2

• if z=x+iy then |e^z| = e^x = e^{Re(z)}
• ∀z,w ∈C, e^{z+w} = e^z • e^w. (Sum powers if product)
• the exponential functi n e^z is PERIODIC with period 2pii

• ∀z ∈C. Exp(z) not equal to 0 and
1/ e^z = e^-z

Remember the real part of z is not equal to the modulus of z and in the exponential we have:
for all z in complex plane , e^z ≠0, 1/e^z = e^-z

Question 3

Cosh z

Answer 3

Even function

Cosh z = 0.5( e^z + e^-z)

Defined on the complex plane

Question 4

Cos z

Answer 4

Even function cos z = cos-z

Cos z = 0.5( e^iz + e^-iz)

Imaginary exponent power

Defined on the complex plane

Question 5

Sinh z

Answer 5

Odd function

Sinh z = 0.5( e^z - e^-z)

Defined on the complex plane

Question 6

Sin z

Answer 6

Odd function
Sin z = -sin z

Sin z = (1/2i) •( e^iz -e^-i z)

Defined on the complex plane

Question 7

Relationships between cos sin sinh and cosh

Answer 7

• cos z = cosh(iz)
• sin (z+w) = sinz•cosw + cosz• sinw
• isinZ = sinh(iz)
• sin ² z + cos ² z = 1

This does not imply |sinz| ≤ 1. & |cosz| ≤ 1 for all complex z
(In fact sin and cos are unbounded on C)

Question 8

Eg z=iy

|cosz| =
For y in reals

Answer 8

|cosz| = |cosiy| = |coshy| = coshy which tends to infinity as y tends to infinity

Question 9

Example find M st

For all z with |z| =1

(e^z +cosz) /(6+z)| ≤ M

Answer 9

Let z=x+iy then

|cosz| = | 0.5(e^iz + e^-iz)|

By the triangle inequality

≤ 0.5|e^iz| + 0.5|e^-iz|

= 0.5 e^(Re(iz)) + 0.5 e^ (Re(-iz))

= 0.5 e^-y + 0.5 e^y = coshy

By definition of coshy

Hence for all z with |z|=1
|cosz| ≤ coshy ≤cosh1 and the triangle rule 
|e^z + cosz| ≤ |e^z| + coshy
≤ e^Re(z) + coshy
≤ e^1 +cosh1

|6+z| ≥ |6| -|z| =5 giving M = (e +cosh1)/(5)

Question 10

Find the 0s of cosz

Answer 10

Let z = x+iy

(Remember z is in the complex plane so we have more roots)

Cosz = cos( x+iy)
= cos x cosiy - sinx siniy

By definition
= cosx (coshy) -i sinx(sinhy)

Taking the modulus and squaring:
|cosz| ² = cos ²x cosh ²y + sin ²x sinh ²y

(Note it’s a+ not - because the modulus doesn’t involve squaring I’s only the y part)

= cos ²x + sinh ²y
So we have
cos ²x + sinh ²y =0 ie (cosx-sinhy)(cosx +sinhy)=0

Cosx= -sinhy ——>
cos x =sinhy =0

Since x and y in reals
x= pi/2 + n*pi And y=0.

Thus solutions are
Z= pi/2 + n*pi (n=0, ±1, ±2,…) which are on the real axis

Question 11

Tan z.

Answer 11

Excluding points at which the denominator is 0

Tan z = sinz /cosz

Except for z is (2n+1) π /2

arctan for arguement could be plus pi!!

Question 12

Sec z

Answer 12

Excluding points at which the denominator is 0

Secz = 1 /cosz

Except for z is (2n+1) π /2

Question 13

Tanhz

Answer 13

Excluding points at which the denominator is 0

Tanh z = sinhz /coshz

Except for z is (2n+1) πi /2

Question 14

Cot z

Answer 14

Excluding points at which the denominator is 0

Cot z = cosz /sinz

Except for z is nπ

For integers n

Question 15

Cosec z

Answer 15

Excluding points at which the denominator is 0

Cosec z= 1/sinz

Except for z is nπ

For integers n

Question 16

Coth z

Answer 16

Excluding points at which the denominator is 0

Cothz = coshz /sinhz

Except for z is nπi

Question 17

Cosech z

Answer 17

Excluding points at which the denominator is 0

Cosech z = 1/ sinhz

Except for z is nπi

Question 18

properties Log Z

Answer 18

If log z has a value Then it has an infinite number of values and log z is not a function

•e^x is a strictly increasing positive function on R, define inverse function lnx on (0, ∞)

• this for all real x e^lnx =x
• if we suppose z is not equal to 0 any w in C st e^w = z is value of log z then for all n in Z

exp( w + 2nπi) = exp{w} = z

So if w is a value of log z so is w+ 2nπi is also, due to periodicity

log z is defined on C{0}

Question 19

Example:

FInd all values of log(-e)

Answer 19

mod arg form: -e = e(cos π + isin π)

so a value of log(-e) is ln | − e| + i arg(−e) = 1 + iπ

All values are therefore given by:

1 + i(π + 2nπ) (n ∈ Z)

Question 20

Example:

FInd all values of log( √3 − i)

Answer 20

mod arg form: √3 − i = 2(cos (-π/6) + isin(- π/6))

so a value of log( √3 − i) is
ln |2| + i arg(√3 − i) = ln 2 − iπ/6

All values are therefore given by:

ln2 - i(π/6) + (2nπi) (n ∈ Z)

Question 21

VALUES OF LOG Z

complex logarithm

Answer 21

,if z ≠ 0 and z = reiθ with r bigger than 0, then

log z = ln r + iθ + 2nπi
= ln |z| + i arg z,
where arg z has infinitely many values, so that log z also has infinitely many values and
any pair differ by an integral multiple of 2πi

log z is not defined when z = 0

Question 22

Find all roots of the equation cosh z = −i.

Answer 22

cosh z =(1/2)(e^z + e^−z)

so equation becomes
eᶻ + e⁻ᶻ = 2i
e^(2z) + 2ie^z + 1 = 0
quadratic eq with roots

eᶻ = -i ±√ ̅ ̅i2̅ ̅−̅ ̅1̅ = -i ± √2
by the quadratic formula

Thus required values of z are the values of log(−i ± i√2).

log(−i − i√2) has values ln(√2 + 1) −πi/2 + 2nπi (n ∈ Z),

log(−i + i√2) has values ln(√2 - 1) +πi/2 + 2nπi (n ∈ Z),
(ln(√2 − 1) = − ln(√2 + 1) )

REQUIRED SOLS ARE
z = ±[ln(√2 + 1) −πi/2 + 2nπi]

(n ∈ Z).

Question 23

cosh^2y-sinh^2y

Answer 23

=1

Question 24

EXAMPLE: Find all roots of e^2z +1+i =0

Answer 24

e^2z = -1-i

so values log(-1-i) are
ln√2 - 3πi/4 +2nπi n in Z

so z= 0.5 ln√2 - 3πi/8 + nπi n in Z

Question 25

log z lnz ln|z|

Answer 25

logz is complex log

ln|z| is ln of a positive real