This is an online educational tools of Complex Graphing calculator to create graph of mathematical complex numbers or functions of 2D in this section.There are three portions in this section. For First portion, you can select any of RealNum, Mod,Arg,Re, Im which are real number, modulus, argument, real part, imaginary part of complex number of 1st input box. then select any one of +, -, < and then you can done similarly for Second and Third portions. then the graph will drawn automatically. It takes sometimes to draw graph of this function and be patience to calculate huge points to draw the graph. For zoom this graph, you can change the x-scale and y-scale. It can draw any complex number or function of equation or inequality.It has some limitations such as you can type with z, z^2, z^3, 1/z, conjz and any complex number e.g. 5+3i.Here conjz means conjugate number of z.Thank you for viewing this article.
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Imaginary Number : The square root of a negative real number is called Imaginary Number. The Unit of imaginary number is denoted by i=√(-1). As for example, √(-9)=3i.
Complex Number : The Complex Number is defined by a+ib where a and b are real numbers and i=√(-1). As for example, 3+5i is a Complex Number.
Conjugate Complex Number : If z=x+iy be a Complex Number, then z=x-iy is called the Conjugate Complex Number of complex number z. It is denoted by z=x-iy. As for example, If z=5+4i, then z=5-4i and If z=-3-6i, then z=-3+6i.
Modulas : If z=x+iy be a Complex Number and x=rcosθ, y=rsinθ, then r=|z|=√(x2+y2) is called the modulas of z. As for example, If z=3+4i, then Modulas |z|=√(32+42)=5.
Argument : If z=x+iy be a Complex Number and x=rcosθ, y=rsinθ, then θ=tan-1(y/x) is called the Argument of z. As for example, If z=1+i, then Argument θ=tan-1(1/1)=450.
Real part : If z=x+iy be a Complex Number, then x is called the Real part of complex number z. It is denoted by Re(z)=x. As for example, If z=7+4i, then Re(z)=7.
Imaginary part : If z=x+iy be a Complex Number, then x is called the Imaginary part of complex number z. It is denoted by Im(z)=y. As for example, If z=7+4i, then Im(z)=4.
Argand plane : The set C of complex numbers is naturally identified with the plane R2. This plane is called the Argand plane. A complex number z = x+i y, its real and imaginary parts x and y define an element (x, y) of R2.
Reciprocal of a Complex Number : The reciprocal of a complex number is a complex number which is calculating 1 divided by that complex number. For example, the reciprocal of a complex number 1+i is 1/(1+i)=(1-i)/{(1+i)(1-i)}=(1-i)/(1-i2)=(1-i)/{(1-(-1)}=(1-i)/2=1/2-i/2
The equation of ellipse of complex number system is |z-z1|+|z-z2|=constant. As for example, |z+3+8i|+|z-3-8i|=18 is an equation of ellipse of complex number. For learn about simple function of Mathematics and graphing, visit: Graphing simple function of Mathematics
The equation of Hyperbola of complex number system is |z-z1|-|z-z2|=constant. As for example, |z+5i|-|z-5i|=12 is an equation of hyperbola of complex number. For learn about piecewise function and graphing, visit: Graphing piecewise function of Mathematics
Another equation of Hyperbola of complex number system are Re(z2)=constant and Im(z2)=constant . As for example, Im(z2)=15 is an equation of hyperbola of complex number. For learn about implicit function and graphing, visit: Graphing Implicit function of Mathematics
The equation of Circle of complex number system is |z-z1|=constant. As for example, |z-5-2i|=6 is an equation of circle of complex number. For learn about linear programmming of Mathematics and graphing, visit: Graphing linear programmming of Mathematics
The inequality of ring shape region of complex variable 3<=|z+3i|<=5 is given bellow and.
Youtube vedio tutorial for complex analysis.
For learn about inequality of Mathematics and graphing, visit: Graphing Inequality of Mathematics
The inequality of outer region of hyperbola of complex variable Re(z2)>2 is given bellow-
For learn about polar function of Mathematics and graphing, visit: Graphing Polar function of Mathematics
The inequality of outer region of hyperbola of complex variable Im(z2)>2 is given bellow-
For learn about parametric function of Mathematics and graphing, visit: Graphing Parametric function of Mathematics
Definition of Comlex analysis
The Complex analysis is the study of analysis of Complex Numbers together with their graphs, derivatives,integrations, manipulation and other mathematical properties. Complex analysis is also known as one of the classical branch of mathematics and analysis. The Complex Numbers with their functions and their graphs, limits, derivatives, integrations, manipulation and other mathematical properties. The
The Complex analysis is a mathematical topic of practical applications to solve physical problems. The real life applications of complex analysis are Signal processing, AC circuit analysis,
The Quantum mechanics,
The Electronics,
The Electromagnetism,used in
The computer science engineering,used in mechanical and
The civil engineering, used in
The control systems, etc.
Euler's formula or Euler's theorem of Complex analysis:
If i is the unit of imaginary number and where e is the base of the natural logarithm, Then the Euler's formula or Euler's theorem of Complex analysis is defined by eiθ=cosθ+isinθ where θ is a angle. The Euler's formula or Euler's theorem , named from Swiss Mathematician Leonhard Euler. The Euler's formula or Euler's theorem of Complex analysis establishes the fundamental relationship between the complex exponential function and the trigonometric functions i.e. eix;=cosx+isinx where e is the base of the natural logarithm and i is the unit of imaginary number.
De Moiver’s Theorem of Complex analysis:
If i is the unit of imaginary number, Then the De Moiver’s Theorem of Comlex analysis is defined by (cosx+isinx)n=cosnx+isinnx where x is a angle and n is any integer. The theorem , named from The Mathematician Abraham de Moivre.
The relation between Euler's formula and De Moiver’s Theorem of Complex analysis is De Moivre's formula is a precursor to Euler's formula. Anyone can derive de Moivre's formula by applying the Euler's formula and the exponential law of integer i.e. (eix)n=einx=cosnx+isinnx .
:If z1 and z2 are two complex numbers and |z1|,|z2| and |z1 + z2| are modulus of z1, z2 and z1+z2 respectively, then the Triangle Inequality of complex numbers is |z1 + z2|≤|z1| + |z2|. This theorem is generalized by |z1 + z2 + z3+....+zn|≤|z1| + |z2|+|z3|+.....+|zn|.
Parametric equation of a circle of complex number on complex plane is z(t)=rcost+irsint=reit where 0≤t≤2π and r is the radius of the circle. As for example,z(t)=10cost+10isint=10eit where 0≤t≤2π is a circle of radius 10 units and the center is origin. The general equation of a circle is x2+y2+2gx+2fy+c=0 where the center is (-g, -f), so the parametric general equation of the circle on complex plane will be z(t)=(-g+rcost)+i(-f+rsint).
The general equation of an ellipse is x2/a2+y2/b2=1 where the center is (0, 0), so the parametric equation of the ellipse on complex plane will be z(t)=(acost)+i(bsint) where 0≤t≤2π.
The general equation of a hyperbola is x2/a2-y2/b2=1 where the center is (0, 0), so the parametric equation of the hyperbola on complex plane will be z(t)=(asect)+i(btant) where 0≤t≤2π.
The general equation of a parabola is y2=4ax, so the parametric equation of the parabola on complex plane will be z(t)=(at2)+i(2at) where 0≤t≤2π.
N.B: If you want to drawing graph of parametric equation on complex plane, you will use graphing parametric calculator on complex plane.
A complex equation has an asymptote if the distance between a straight line and the curve of complex equation tends to zero when the points on the curve approach to infinity, then the straight line is called asymptote. As for example the equation of complex variable Re(z2)=15 ⇒ Re(x2-y2+2xyi)=15 ⇒ x2-y2=15. Now we can find the equation of asymptote, by seting x2-y2=0 ⇒(x+y)(x-y)=0 ⇒x+y=0 or x-y=0 ⇒y=-x, y=x . The graph of the equation of complex variable Re(x2-y2+2xyi)=15 and its asymptotes y=x, y=-x are given bellows-
Single valued function: A complex function w is said to be a single valued function of z if it takes only one value for each value of z. As for example, w=f(z)=z2 is a single valued functtion. Suppose f(1+i)= (1+i)2=1+2i+i2=1+2i-1=2i which is single value.
A function w=f(z) is said to be limit l as z tends to z0 along any path in a defined region and we write limz→z0f(z)=l if for any positive number ε (however small) there exists a positive number δ (depending on ε) such that |f(z)-l|<ε whenever 0<|z-z0|<δ i.e. ∃ a deleted neighbourhood of the point z=z0 in which |f(z)-l|<ε can be made as small as we please.
A complex function w=f(z) is said to be continuous at a point z0 if (i) f(z0) is defined (ii) limz→z0f(z) exists (iii) limz→z0f(z)=f(z0).
Another Defination: A complex function w=f(z) is said to be continuous at a point z0 if for given ε>0, there exists a number δ>0 (depending on ε and z0) such that |f(z)-f(z0)|< ε whenever |z-z0|< δ.
Continuous complex function: A complex function w=f(z) defined in a certain domain D is said to be a continuous function on D , if it is continuous at each
point of its domain D.
Let w=f(z) be a complex function defined at all points in same neighbourhood of a point z0. The derivative or differential coefficient
of f(z) at z0 is written as f'(z) and is defined by f(z0)=limz→z0
{f(z)-f(z0}/(z-z0) provided that
the limit exists. f(z) is said to be differentiable at z0 when its derivative at z0 exists. The process of evaluating f'(z0) is called
differentiation or derivation.
A complex function w=f(z) is said to be analytic or holomorphic at a point z0 if there exists a neighbourhood |z-z0|<δ
at all points of which f'(z) exists.
If a function f(z) fails to be analytic at a point z0 but is analytic at some points in every neighbourhood of z0, then
z0 is called a singular point or singularity of z. As for example, f(z)=1/z, then f(z) is analytic except at z=0 . So z=0 is a singular point of f(z)=1/z.
Harmonic function: Any real valued function H of two variables of x and y having continuous partial derivatives of first and second order in a region R and also
satisfying Laplace's equation δ2H/δx2+δ2H/δy2=0
i.e. Δ2H=0 is called a harmonic function.
As for example, u=x3-3xy2 so δu/δx=3x2-3y2
and δu/δy=-6xy
that implies δ2H/δx2=6x and δ2H/δy2=-6x
Hence δ2H/δx2+δ2H/δy2=6x-6x=0 which shows that u satisfies the Laplace's equation . Hence u is a harmonic function.
Branch of multiple valued function: A multiple valued function can be considered as a collection of single valued function, each numbers of which is called
a branch of the function. Each of the branches of a multiple valued function is a single valued function and is analytic.
Cauchy riemann partial differential equations in Cartesian form : If f(z)=u(x,y)+iv(x,y) is an analytic function, then the cauchy riemann partial differential equations
in Cartesian form are δu/δx=δv/δy
and δu/δy=-δv/δx i.e. ux=vy and uy=-vy.
Laplace partial differential equation in cartesian form: Laplace partial differential equation of φ is δ2φ/δx2+δ2φ/δy2=0
i.e. Δ2φ=0 .
Let f(z) is integrable along the curve C having finite length L and if there exists a positive number M such that |f(z)|≤M on the curve C, then |∫C f(z)dz|≤ |∫C f(z)||dz| and |∫C f(z)dz|≤ML.
Let f(z) is analytic and f'(z) is continuous at all points inside and on a simple closed curve C, then ∫C f(z)dz=0.
Let f(z) is continuous in a simple connected region R and if ∫C f(z)dz=0 around every simple closed curve C in R, then f(z) is analytic in R.
In 1900 the French mathematician Edouard Goursat proved that the assumption of continuity is not necessary to reach the conclusion of Cauchy's theorem. The resulting modified version of Cauchy's theorem is known today as the Cauchy-Goursat Theorem. Cauchy's fundamental theorem was first provided by use of Green's theorem with the added restriction that f'(z) should be continuous in a region R but Goursat showed that for the truth of the theorem the assumption of the continuity of f'(z) is unnecessary and that cauchy's theorem holds if and only if f(z) is analytic in a region R and on its boundary C. Thetheorem is also known as General form of cauchy's theorem.
Let f(z) is analytic inside and on a simple closed curve C of a simple connected region R and a be any point inside C, then f(a)=1/2πi∮ c f(z)/(z-a)dz, where C is traversed in the positive sense.
Let f(z) is analytic inside and on a simple closed curve C of a simple connected region R and a be any point inside C, then f'(a)=1/2πi∮ c f(z)/(z-a)2dz, where C is traversed in the positive sense.
Let f(z) is analytic inside and on a simple closed curve C of a simple connected region R and a be any point inside C, then fn(a)=n!/2πi∮ c f(z)/(z-a)n+1dz, where C is traversed in the positive sense.
f(z) is analytic inside and on a circle C of radius r and center at z=a,then |fn(a)|≤M.n!/rn for n=0, 1, 2, .... Where M is a constant such that |f(z)|<M on C i.e. M is an upper bound of |f(z)| on C.
Taylor's Theorem or Taylor's series: If f(z) is analytic inside a circle C with center at a, then for all z inside C
f(z)=f(a)+f'(a).(z-a)/1!+f''(a).(z-a)2/2!+f'''(a).(z-a)3/3!+... ... ...+f(n)(a).(z-a)n/n!+... ... to ∞
Maclaurin Theorem or Maclaurin Series: If f(z) is analytic inside a circle C with center at 0, then for all z inside C
f(z)=f(0)+f'(0).z/1!+f''(0).z2/2!+f'''(0).z3/3!+... ... ...+f(n)(0).zn/n!+... ... to ∞
Laurent Theorem or Laurent series: If f(z) is analytic inside and on the boundary of the ring-shaped region R bounded by the circles C
with center at c and the respective radius r, then for all z in R
Zeros of a polynomial or Roots of a polynomial equation: Let us consider a polynomial equation a0zn+a1zn-1+a2zn-2+... ...+an-1z+an=0....(i)
where a0≠ 0,a1, a2, ... ... an are complex numbers and n is a positive integer which is called the degree of the equation. Then the solutions
of (i) are called the roots of the polynomial equation or the zeros of the polynomials a0zn+a1zn-1+a2zn-2+... ...+an-1z+an.
Ordinary point in complex analysis: Let f(z) is a complex function. If z0 is not a singular point and if there exist a δ>0 such that the circle |z-z0|=δ
encloses no singular point, then z0 is called an ordinary point of a function f(z).