complex function ,separate into real and imaginary parts 1.exp(z^2),z=(x+iy) 2. exp(5+iπ/2) 3. e^(16+30i)

                                        BS GREWAL SOLUTION

   Complex functions

If for each value of the complex variable Z(x+iy) in a given region R,we have one or more values of W(-u+iv) ,then w is said to be a complex number of Z

Written as f(z) = w = u(x,y)+iv(x,y)

Exponential function of a complex variable

e^x= 1+x/1!+x^2/2!+ ---------+x^n/n!+-----------,x €R

then e^z=1+z/1!+z^2/2!+ ---------+z^n/n!+---------, z € Ө

properties:-

* Exponential form of z =r(e^iӨ)

*e^z is periodic function having imaginary period 2πi

.i.e. e^(z+2πi) =e^z* e^2πi = e^z

·         e^z is not zero for any value of z

·      e^z = e^z⎺

 

question:-separate into real and imaginary parts

1.exp(z^2),z=(x+iy)

2. exp(5+iπ/2)

3. e^(16+30i)

HERE IS THE SOLUTION

Solution   1. :-   e^(z^2)=e^(x+iy)^2

 e^(x^2-y^2+2ixy) = e^(x^2-y^2)*e^(2ixy)

(e^(x^2-y^2))*(cos2xy+isin2xy)

 real part= e^(x^2-y^2)

imaginary part= e^(cos2xy+isin2xy)

 

Solution   2.:- exp(5+iπ/2)=e^(5+iπ/2)

Let x+iy=e^5*e^ iπ/2

             =  e^5*(cosiπ/2+isin π/2)

=e^5*(i)

y= e^5*(i)

 

Solution   3.:- e^(16+30i)

X=16 and y=30

 

e^16*(cos30+isin30)

e^16*(√ 3/2+i1/2)

solve (x^2 - 4xy -2y^2)dx+(y^2-4xy -2x^2)dy=0