Question: Prove that f(z) = e^z is differentiable at any point z
Question: Prove that f(z) = e^z is differentiable at any point z Solution: Given complex values function f(z) = e^z Put z =x+iy f(z) = e^z=e^(x+iy)=e^x e^(iy) f(z) = e^z…
solved Show that f(z) = |z|^2 is differentiable only at z=0
Question : Show that f(z) = |z|^2 is differentiable only at z=0 Solution : Given complex function f(z) = |z|^2 = z z ̅ Since z= x+iy and z ̅= x-iy, i^2=-1…
Show that f(z)=conjugate(z) has no derivative
Question : Show that f(z)=conjugate(z) has no derivative Solution : We know that a complex conjugate finction is defined as f(z)= z ̅=x-iy Then Re f(z) = u(x, y) = x…
how do you find real and imaginary parts of cos(z)
Question : Find real and imaginary parts of cos(z) Solution : First we shall put \[z=x+iy\] then \[cos(z)=cos(x+iy)\]now using the formula of \[cos(A+B)=cos(A)cos(B)-sin(A)sin(B)\] we get \[cos(x+iy) = cos(x)cos(iy)-sin(x)sin(iy)\] now by…
how to find real and imaginary parts of e^(x+iy)
Question : Find real and imaginary parts of e^(x+iy) Solution : We know that Real and imaginary parts of complex Number z = a+ib are a and b respectively.So we…
what is dimension of the vector space C(R) of the complex numbers over real
Question : What is The dimension of the vector space C(R) of the complex numbers over real numbers ? Solution : We know that a complex number z is written…
how to Find sinh(x+iy) formula
Question : Find sinh(x+iy) formula Solution : Let the complex number as \[z=sinh\left(x+iy\right)\] We know that \[\sin{\left(iz\right)}=isinh\left(z\right)\] So we shall multiply and divide by i we get \[z=\frac{isinh\left(x+iy\right)}{i}=\frac{\sin{\left(i\left(x+iy\right)\right)}}{i}=\frac{\sin{\left(ix-y\right)}}{i}\] Now we…
real and imaginary part of e^(sin(x+iy))
Question : Find the real and imaginary part of e^(sin(x+iy)) Solution: Let \[z = e^(sin(x+iy))\] We shall write it as \[z = e^{Sin\left(x+iy\right)}=e^{Sin\left(x\right)Cos\left(iy\right)+\cos{\left(x\right)}\sin{\left(iy\right)}}\] Since \[Cos\left(iy\right)=\cosh{\left(y\right)}\ ,\ \ \sin{\left(iy\right)}=isinh\left(y\right)\] we shall…
What is a complex number ?
Introduction : A complex number is a number which contains real and imaginary parts. Complex numbers are very useful in every fields of study. Consider an equation x^2+1=0 which is…
real and imaginary part of log(z)
Introduction : Here we shall find real and imaginary part of log(z). Real and imaginary parts of sin(z) are obtained by expanding this function. Question : Find real and imaginary…