Mister Exam
(log(y)+1)/y=c-log(x) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
log(y) + 1
---------- = c - log(x)
y
$$\frac{\log{\left(y \right)} + 1}{y} = c - \log{\left(x \right)}$$
Detail solution
Given the equation
$$\frac{\log{\left(y \right)} + 1}{y} = c - \log{\left(x \right)}$$
Transfer the right side of the equation left part with negative sign
$$\log{\left(x \right)} = c - \frac{\log{\left(y \right)} + 1}{y}$$
This equation is of the form:
By definition log
then
$$x = e^{\frac{c - \frac{\log{\left(y \right)} + 1}{y}}{1}}$$
simplify
$$x = e^{c - \frac{\log{\left(y \right)} + 1}{y}}$$
$$\frac{\log{\left(y \right)} + 1}{y} = c - \log{\left(x \right)}$$
Transfer the right side of the equation left part with negative sign
$$\log{\left(x \right)} = c - \frac{\log{\left(y \right)} + 1}{y}$$
This equation is of the form:
log(v)=p
By definition log
v=e^p
then
$$x = e^{\frac{c - \frac{\log{\left(y \right)} + 1}{y}}{1}}$$
simplify
$$x = e^{c - \frac{\log{\left(y \right)} + 1}{y}}$$
The graph
Rapid solution
[src]
re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y) re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y)
- --------------- + --------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- - --------------- - ---------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
/ im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \ im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) / im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \
x1 = cos|--------------- + --------------- + --------------- - --------------- - ---------------|*e + I*e *sin|--------------- + --------------- + --------------- - --------------- - ---------------|
| 2 2 2 2 2 2 2 2 2 2 | | 2 2 2 2 2 2 2 2 2 2 |
\im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/ \im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/
$$x_{1} = i e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \sin{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)} + e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \cos{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}$$
x1 = i*exp(-log(|y|)*re(y)/(re(y)^2 + im(y)^2) + re(y)*re(c*y)/(re(y)^2 + im(y)^2) - re(y)/(re(y)^2 + im(y)^2) + im(y)*im(c*y)/(re(y)^2 + im(y)^2) - im(y)*arg(y)/(re(y)^2 + im(y)^2))*sin(log(|y|)*im(y)/(re(y)^2 + im(y)^2) + re(y)*im(c*y)/(re(y)^2 + im(y)^2) - re(y)*arg(y)/(re(y)^2 + im(y)^2) - re(c*y)*im(y)/(re(y)^2 + im(y)^2) + im(y)/(re(y)^2 + im(y)^2)) + exp(-log(|y|)*re(y)/(re(y)^2 + im(y)^2) + re(y)*re(c*y)/(re(y)^2 + im(y)^2) - re(y)/(re(y)^2 + im(y)^2) + im(y)*im(c*y)/(re(y)^2 + im(y)^2) - im(y)*arg(y)/(re(y)^2 + im(y)^2))*cos(log(|y|)*im(y)/(re(y)^2 + im(y)^2) + re(y)*im(c*y)/(re(y)^2 + im(y)^2) - re(y)*arg(y)/(re(y)^2 + im(y)^2) - re(c*y)*im(y)/(re(y)^2 + im(y)^2) + im(y)/(re(y)^2 + im(y)^2))
Sum and product of roots
[src]
sum
re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y) re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y)
- --------------- + --------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- - --------------- - ---------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
/ im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \ im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) / im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \
cos|--------------- + --------------- + --------------- - --------------- - ---------------|*e + I*e *sin|--------------- + --------------- + --------------- - --------------- - ---------------|
| 2 2 2 2 2 2 2 2 2 2 | | 2 2 2 2 2 2 2 2 2 2 |
\im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/ \im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/
$$i e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \sin{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)} + e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \cos{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}$$
=
re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y) re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y)
- --------------- + --------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- - --------------- - ---------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
/ im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \ im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) / im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \
cos|--------------- + --------------- + --------------- - --------------- - ---------------|*e + I*e *sin|--------------- + --------------- + --------------- - --------------- - ---------------|
| 2 2 2 2 2 2 2 2 2 2 | | 2 2 2 2 2 2 2 2 2 2 |
\im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/ \im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/
$$i e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \sin{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)} + e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \cos{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}$$
product
re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y) re(y) im(y)*im(c*y) re(y)*re(c*y) arg(y)*im(y) log(|y|)*re(y)
- --------------- + --------------- + --------------- - --------------- - --------------- - --------------- + --------------- + --------------- - --------------- - ---------------
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
/ im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \ im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) / im(y) im(y)*log(|y|) im(c*y)*re(y) arg(y)*re(y) im(y)*re(c*y) \
cos|--------------- + --------------- + --------------- - --------------- - ---------------|*e + I*e *sin|--------------- + --------------- + --------------- - --------------- - ---------------|
| 2 2 2 2 2 2 2 2 2 2 | | 2 2 2 2 2 2 2 2 2 2 |
\im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/ \im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y) im (y) + re (y)/
$$i e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \sin{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)} + e^{- \frac{\log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}} \cos{\left(\frac{\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{\operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{\operatorname{im}{\left(y\right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} \right)}$$
=
-re(y) + im(y)*im(c*y) + re(y)*re(c*y) - arg(y)*im(y) - log(|y|)*re(y) I*(im(y)*log(|y|) + im(c*y)*re(y) - arg(y)*re(y) - im(y)*re(c*y) + im(y))
---------------------------------------------------------------------- + -------------------------------------------------------------------------
2 2 2 2
im (y) + re (y) im (y) + re (y)
e
$$e^{\frac{- \log{\left(\left|{y}\right| \right)} \operatorname{re}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{re}{\left(c y\right)} - \operatorname{re}{\left(y\right)} + \operatorname{im}{\left(y\right)} \operatorname{im}{\left(c y\right)} - \operatorname{im}{\left(y\right)} \arg{\left(y \right)}}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{i \left(\log{\left(\left|{y}\right| \right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} \operatorname{im}{\left(c y\right)} - \operatorname{re}{\left(y\right)} \arg{\left(y \right)} - \operatorname{re}{\left(c y\right)} \operatorname{im}{\left(y\right)} + \operatorname{im}{\left(y\right)}\right)}{\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}}}$$
exp((-re(y) + im(y)*im(c*y) + re(y)*re(c*y) - arg(y)*im(y) - log(|y|)*re(y))/(im(y)^2 + re(y)^2) + i*(im(y)*log(|y|) + im(c*y)*re(y) - arg(y)*re(y) - im(y)*re(c*y) + im(y))/(im(y)^2 + re(y)^2))