Mathematics for Applied Sciences (Osnabrück 2011-2012)/Part I/Exercise sheet 3



Warm-up-exercises

Show that the binomial coefficients satisfy the following recursive relation

 


Show that the binomial coefficients are natural numbers.


Prove the formula

 


Show by induction that for   the estimate

 

holds.


In the following computing tasks regarding complex numbers, the result must always be given in the form  , with real numbers  , and these should be as simple as possible.

Calculate the following expressions in the complex numbers.

  1.  .
  2.  .
  3.  .
  4.  .
  5.  .
  6.  .


Show that the complex numbers constitute a field.


Prove the following statements concerning the real and imaginary parts of a complex number.

  1.  .
  2.  .
  3.  .
  4. For   we have
     
  5. The equation   holds if and only if   and this holds if and only if  .


Prove the following calculating rules for the complex numbers.

  1.  .
  2.  .
  3.  .
  4.  .
  5. For   we have  .


Prove the following properties of the absolute value of a complex number.

  1.  
  2. For a real number   its real absolute value and its complex absolute value coincide.
  3. We have   if and only if  .
  4.  
  5.  
  6. For   we have  .
  7.  


Check the formula we gave in example for the square root of a complex number

 

in the case  .


Determine the two complex solutions of the equation

 




Hand-in-exercises

Exercise (3 marks)

Prove the following formula

 


Exercise (3 marks)

Calculate the complex numbers

 

for  .


Exercise (3 marks)

Prove the following properties of the complex conjugation.

  1.  .
  2.  .
  3.  .
  4. For   we have  .
  5.  .
  6.   if and only if  .


Exercise (2 marks)

Let   with  . Show that the equation

 

has at least one complex solution  .


Exercise * (5 marks)

Calculate the square roots, the fourth roots and the eighth roots of  .


Exercise (3 marks)

Find the three complex numbers   such that