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# Oxford prep. questions help. watch

1. i dont get questions 6.6 and 6.7, for those that have the booklet...otherwise:

6.6)find all the complex roots of
a) cosh z =1
b)sinh z= 1
c)e^z = -1
d)cos z=sqrt 2

6.7) Show:
w=z+c/z, maps circle |z|=1 uin the z plane into an ellipse in the w plane and find its eqtn

spent 15 mins or so on each question and gave up
cheers

phil
2. 6.6)find all the complex roots of
a) cosh z =1
cosh(a+ib)=1
cosha.coshib+sinha.sinhib=1
cosha.cosb+isinha.sinb=1
Clearly:
cosha.cosb=1 and sinha.sinb=0
If sinha=0 then a=0 so cosha=1 and so cosb=1 giving b=2kpi.
If sinb=0 then b=kpi and cosb=+-1 but we require cosb to be positive as cosha is positive so b=2kpi and a=0.
Hence the complex roots are 0+i(2kpi) where

b)sinh z= 1
Very similar to the above but using the correct identity for sinh(A+B).

c)e^z = -1
cosz+isinz=-1
Clearly cosz=-1 and sinz=0.
Hence z=(2k+1)pi where

d)cos z=sqrt 2
Use z=a+ib, expand and equate real and imaginary coefficients as in (a) or (b). Obviously remember to use the correct identities as you're dealing with a trigonometric rather than hyperbolic function.

6.7) Show:
w=z+c/z, maps circle |z|=1 uin the z plane into an ellipse in the w plane and find its eqtn
Clearly attempts to rearrange the formula to find z in terms of w are going to give us a quadratic and thus things will get a little messy.
w=x+iy+c/(x+iy)
w=x+iy+c(x-iy)/(x^2+y^2)
w=x+iy+c(x-iy)/1
w=x+iy+cx-icy
w=(x+cx)+i(y-cy)
w=x(c+1)+iy(1-c).
If w=u+iv then u=x(c+1), v=y(1-c).
Clearly x=u/(c+1) and y=v/(1-c).
You know x^2+y^2=1 so u^2/(c+1)^2+v^2/(1-c)^2=1.
c is a constant so u^2/(1+c)^2+v^2/(1-c)^2=1 is of the form u^2/a^2+v^2/b^2=1 where a and b are constants so the locus is indeed an ellipse.
3. thanks for that!
4. (Original post by Phil23)
thanks for that!
No problem but I made a minor correction. Sorry about the confusion.
5. i dont get the english of this problem:

A circular cone has its vertex at the origin and its axis in the direction of the unit vector a. The half angle at the vertex is alpha.

what is a hlaf angle? - is it the angle that encompasses the two sides of the cone or is it half of this...i.e. the angle between the axis of symmetry and one side?
6. (Original post by Phil23)
i dont get the english of this problem:

A circular cone has its vertex at the origin and its axis in the direction of the unit vector a. The half angle at the vertex is alpha.

what is a hlaf angle? - is it the angle that encompasses the two sides of the cone or is it half of this...i.e. the angle between the axis of symmetry and one side?
It's that

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