Free Onlne graphing calculator on complex analysis or complex functions or complex equation or complex inequality

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

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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

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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

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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

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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

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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

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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

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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.

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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 e^iθ=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. e^ix;=cosx+isinx where e is the base of the natural logarithm and i is the unit of imaginary number.

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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. (e^ix)ⁿ=e^inx=cosnx+isinnx .

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: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=re^it where 0≤t≤2π and r is the radius of the circle. As for example,z(t)=10cost+10isint=10e^it 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.

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Many valued function or Multiple valued function: A complex function w is said to be a Many valued function or Multiple valued function of z if it takes more than one value for each value of z.As for example, w=f(z)=z^1/3 is a Many valued functtion or multiple valued function. Suppose f(1)= (1)^1/3 which has three values 1, ω and ω².

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 lim_z→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) lim_z→z0f(z) exists (iii) lim_z→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.

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Discontinuous complex function: If a complex function w=f(z) is not continuous at a point z₀ in its domain in the z plane, then w=f(z) is said to be discontinuous at z₀ and z₀ is called a point of discontinuity of f(z).

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Removable discontinuity of a complex function: If lim_z→z0f(z)≠f(z₀),then f(z) is said to have a removable or removal discontinuity at z=z₀ because f(z) can be made continuous by redefining f(z₀) so that lim_z→z0f(z)=f(z₀).

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Uniform continuity of a complex function: A complex function w=f(z) is said to be uniformly continuous in a region R if for given ε>0, it is possible to find a number δ>0 (depending on ε only) such that |f(z₁)-f(z₂)|<ε holds for every pair of points z₁ ,z₂ of the region R for which |z₁-z₂|<δ.

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)=^lim_z→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.

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δ-ε definition: A function f(z) is said to be differentiable at z₀ if for any ε>0, there exists a δ>0 such that |^{{f(z)-f(z₀)}}⁄_(z-z0)-f'(z₀)|<ε whenever 0<|z-z₀|<δ.

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.

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Analytic in a region or Domain: A complex function w=f(z) is said to be analytic in a region R if the derivative f'(z) exists at all points of z of R and is reffered to as an analytic function in R or a function in R.

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.

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Zeros of a Complex Function: A zero of an analytic function f(z) is a value of z for which f(z) vanishes i.e. f(z)=0. Othewise There exist z = z₀ in the domain of f(z) such that f(z₀) = 0, then z₀ is called zero of f(z). As for example, f(z)=zcos(z) has a zero at z=0 of f(z) since f(0)=0.
a function f(z) which is analytic in a domain D, has a zero of order n at point z = z₀ in D if and only if f⁽ⁿ⁾(z₀) = 0. As for example, f(z)=zsin(z) has a zero at z=0 of order 2 since f'(z)=zcos(z)+sin(z) which is equal to 0 at z=0 and
f''(z)=-zsin(z)+cos(z)+cos(z) which is equal to 2 at z=0

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.

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Harmonic conjugate function: If f(z)=u(x,y)+iv(x,y) is a an analytic function, then v(x,y) is called the harmonic conjugate function of u(x,y).

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.

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Branch point in complex analysis: A point z=z₀ is called a branch point of the multiple valued function f(z) if the branches of f(z) are interchanged when z describes a closed path about z₀. otherwise, A branch point of an analytic function is a point in the complex plane whose complex argument can be mapped from a single point in the domain to multiple points in the range. As for example, f(z)=(z-5)^1/2 has a branch point at z=5.

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. u_x=v_y and u_y=-v_y.

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Cauchy riemann partial differential equations in polar form : If f(z)=u(x,y)+iv(x,y) is an analytic function, then the cauchy riemann partial differential equations polar form are ^δu/_δr=(¹/_r)^δv/_δθ and ^δv/_δr=(-¹/_r)^δu/_δθ i.e. u_r=(¹/_r)v_θ and v_r=(-¹/_r)u_θ.

Laplace partial differential equation in cartesian form: Laplace partial differential equation of φ is ^δ2φ/_δx²+^δ2φ/_δy²=0 i.e. Δ²φ=0 .

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Laplace partial differential equation in polar form: Laplace partial differential equation in polar form of φ is ^δ2φ/_δr²+(¹/_r)(^δφ/_δr)+(¹/_r²)^δ2φ/_δθ²=0.