Toronto Math Forum
MAT3342020F => MAT334Tests and Quizzes => Test 2 => Topic started by: Yuyan Liu on October 27, 2020, 06:19:33 PM

Can someone explain how to get to the answer of $cosh(z)$?

$\text{Since we have: } \ cosh(x) = cos(ix) \ \text{ and } \ sinh(x) = i \cdot sin(ix) \\
\text{By substituting and rearranging, we have the following: }$
$
\begin{gather}
\begin{aligned}
f(x,y) &= cosh(x) \cdot sin(y) + (  sinh(x) \cdot cos(y)) \cdot i + C \cdot i \\\\
&= cos(ix) \cdot sin(y)  sin(ix) \cdot cos(y) + C \cdot i \\\\
&= sin(y  ix) + C \cdot i \\\\
&= sin(i \cdot (x+iy)) + C \cdot i \\\\
&=  sin(i(x+iy)) + C \cdot i \\\\
&= i \cdot sinh(x + iy) + C \cdot i \\\\
&= i \cdot sinh(z) + C \cdot i \\\\
\end{aligned}
\end{gather}
$
$\text{I think the answer has a typo, I got} \ i \cdot sinh(z) + C \cdot i \ \text{ instead of } cosh(z) + C \cdot i$

f(x,y) = coshxsiny +i(sinhxcosy+c)
=cos(ix)siny +i((isin(ix)cosy +c)
=cos(ix)siny  sin(ix)cosy +ic
=sin(yix)+ic
=sin(i(x+iy))+ic
=sin(i(x+iy)) + ic
=isinh(x+iy) +ic
=isinhz+ic