CSIR NET MATHEMATICS DEC 2017 | Complex Analysis | Zeros of Complex Function f(z)

Hello friend, this is Mahesh Yolikar and you are watching Next Gen Math. If you have not subscribed our channel, please subscribe our channel and press the bell icon of Next Gen Math. In this video, we are going to discuss the problem of CSI N8, December 2017, Problem of Complex Analysis. Example, the function f c2c defined by f of z equal to e power z plus e power minus z has. We have given four options, first option finitely many zero, second option no zero, third option only real zeros and the fourth option infinitely many zeros. If you want notes of this solution, please visit our website www.nextgenmath.com. In the description link is available. This is a problem of zero of complex function. So first we recall the definition of zero of complex function. A zero of a memomorphic function is a complex number z such that f of z equal to zero. That means zero of complex function is nothing but the complex number z such that f of z equal to zero. So for the given function f of z equal to zero, here f of z equal to e power z plus e power minus z equal to zero. If you write down e power minus z equal to 1 upon e power z, now take the LCM then we have e power 2z plus 1 equal to zero. You can write down e power 2z equal to minus 1, but minus 1 can be written in the form of cos 2n plus 1 pi plus i sin 2n plus 1 pi, n belongs to z. So for cos pi value is minus 1 and sin pi value is zero. So similarly cos 3 pi minus 1 sin 3 pi is zero. So for odd multiple of pi, cos has value minus 1 and sin has value zero, so minus 1 can be written in this form. If you write down in the exponential form then we have e power 2n plus 1 pi i. So if you compare right hand side, left hand side, e power 2z equal to e power 2n plus 1 pi i, then we have 2z equal to 2n plus 1 pi i, n belongs to z. So z equal to 2n plus 1 divided by 2 pi i, n belongs to z. So it is a solution f of z equal to zero. So these are the zeros of this given function f of z equal to e power z plus e power minus z. So how many zeros? So n belongs to z, therefore we have infinitely many zeros. Option number 4 which says that infinitely many zeros, so option number 4 is the correct answer. Thanks for watching. For more videos, like, share, subscribe, press the bell icon of NextGen.